| L(s) = 1 | + 20·5-s − 9·9-s + 92·11-s + 208·19-s + 275·25-s + 448·29-s − 144·31-s + 388·41-s − 180·45-s + 586·49-s + 1.84e3·55-s + 852·59-s − 1.39e3·61-s + 376·71-s − 2.33e3·79-s + 81·81-s − 2.41e3·89-s + 4.16e3·95-s − 828·99-s + 2.25e3·101-s + 3.20e3·109-s + 3.68e3·121-s + 3.00e3·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1/3·9-s + 2.52·11-s + 2.51·19-s + 11/5·25-s + 2.86·29-s − 0.834·31-s + 1.47·41-s − 0.596·45-s + 1.70·49-s + 4.51·55-s + 1.88·59-s − 2.93·61-s + 0.628·71-s − 3.32·79-s + 1/9·81-s − 2.87·89-s + 4.49·95-s − 0.840·99-s + 2.22·101-s + 2.81·109-s + 2.76·121-s + 2.14·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.545270910\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.545270910\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | $C_2$ | \( 1 - 4 p T + p^{3} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 586 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 46 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 3238 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 5470 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 104 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 2562 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 224 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 72 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 100822 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 194 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 147350 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 22754 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 215958 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 426 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 698 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 493942 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 188 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 230434 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 1168 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 973830 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1206 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 90110 T^{2} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.902919696566186985700016688188, −9.359856705317194039034172823374, −8.998086635960919481911835006963, −8.929821366979139158928071203640, −8.420437932161266074418861757186, −7.62817311571429252907055490275, −7.09181458801760521619867815219, −6.96046497984652316455803950632, −6.19671970509778178056173638606, −6.13924826458358244956256096774, −5.60250513752206878391347172866, −5.22474665674629472980698275071, −4.50991321428618044809814007283, −4.19129528310316115391884909280, −3.37420412480245023801702835326, −2.94929778838952601883422259703, −2.44913665195479619226571796935, −1.59800947319570399613448394829, −1.10666592587872645003552225731, −0.923918485388704903815822943164,
0.923918485388704903815822943164, 1.10666592587872645003552225731, 1.59800947319570399613448394829, 2.44913665195479619226571796935, 2.94929778838952601883422259703, 3.37420412480245023801702835326, 4.19129528310316115391884909280, 4.50991321428618044809814007283, 5.22474665674629472980698275071, 5.60250513752206878391347172866, 6.13924826458358244956256096774, 6.19671970509778178056173638606, 6.96046497984652316455803950632, 7.09181458801760521619867815219, 7.62817311571429252907055490275, 8.420437932161266074418861757186, 8.929821366979139158928071203640, 8.998086635960919481911835006963, 9.359856705317194039034172823374, 9.902919696566186985700016688188
