| L(s) = 1 | − 4·5-s + 52·9-s + 80·11-s − 80·19-s + 146·25-s + 280·29-s + 384·31-s − 1.04e3·41-s − 208·45-s + 932·49-s − 320·55-s − 1.39e3·59-s − 1.38e3·61-s + 1.37e3·71-s + 1.47e3·79-s + 954·81-s + 1.32e3·89-s + 320·95-s + 4.16e3·99-s − 1.41e3·101-s + 216·109-s + 1.74e3·121-s − 1.60e3·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 0.357·5-s + 1.92·9-s + 2.19·11-s − 0.965·19-s + 1.16·25-s + 1.79·29-s + 2.22·31-s − 3.99·41-s − 0.689·45-s + 2.71·49-s − 0.784·55-s − 3.07·59-s − 2.90·61-s + 2.30·71-s + 2.09·79-s + 1.30·81-s + 1.57·89-s + 0.345·95-s + 4.22·99-s − 1.39·101-s + 0.189·109-s + 1.31·121-s − 1.14·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.601110849\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.601110849\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + 4 T - 26 p T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 1750 T^{4} - 52 p^{6} T^{6} + p^{12} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 932 T^{2} + 448998 T^{4} - 932 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 40 T + 1526 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 3188 T^{2} + 11156118 T^{4} - 3188 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 2756 T^{2} - 6449082 T^{4} - 2756 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 40 T + 12582 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 228 T^{2} - 31055962 T^{4} - 228 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 140 T + 47534 T^{2} - 140 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 192 T + 13502 T^{2} - 192 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 127124 T^{2} + 8061820662 T^{4} - 127124 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 524 T + 203030 T^{2} + 524 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 227668 T^{2} + 24136249398 T^{4} - 227668 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 208964 T^{2} + 31759440582 T^{4} - 208964 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 29100 T^{2} + 19096738358 T^{4} + 29100 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 696 T + 346006 T^{2} + 696 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 692 T + 554862 T^{2} + 692 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 1150772 T^{2} + 511868382678 T^{4} - 1150772 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 688 T + 809582 T^{2} - 688 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 934628 T^{2} + 433634093478 T^{4} - 934628 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 736 T + 1115358 T^{2} - 736 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 846708 T^{2} + 680383628438 T^{4} - 846708 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 660 T + 1463542 T^{2} - 660 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 3172484 T^{2} + 4151279558022 T^{4} - 3172484 p^{6} T^{6} + p^{12} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.88210507024598909913707012283, −11.40042678631306457050497141872, −10.68580998390805873617843002480, −10.60768452766815369298059392590, −10.49896713264505752614025735804, −9.943478129218277875469814956564, −9.685639651378208100840735886915, −9.328787731590374544541316201445, −8.862201353506884773633177360891, −8.606053448484959275163106501689, −8.395281257672605272275438150788, −7.68293230713065192924247809123, −7.53219239950439294717541199492, −6.91109308591638454018750009951, −6.50567366670184913582507158590, −6.39672957087295814508491037584, −6.35195303421811096072409186746, −5.03601387978227137813381245014, −4.82266239270148470286545803042, −4.45832045849855023244012882801, −3.93459838131123858317257446246, −3.54492742207040441089086225937, −2.72198719258208219626047835126, −1.58543298850250462279469053867, −1.10271753024097661636281778025,
1.10271753024097661636281778025, 1.58543298850250462279469053867, 2.72198719258208219626047835126, 3.54492742207040441089086225937, 3.93459838131123858317257446246, 4.45832045849855023244012882801, 4.82266239270148470286545803042, 5.03601387978227137813381245014, 6.35195303421811096072409186746, 6.39672957087295814508491037584, 6.50567366670184913582507158590, 6.91109308591638454018750009951, 7.53219239950439294717541199492, 7.68293230713065192924247809123, 8.395281257672605272275438150788, 8.606053448484959275163106501689, 8.862201353506884773633177360891, 9.328787731590374544541316201445, 9.685639651378208100840735886915, 9.943478129218277875469814956564, 10.49896713264505752614025735804, 10.60768452766815369298059392590, 10.68580998390805873617843002480, 11.40042678631306457050497141872, 11.88210507024598909913707012283
