| L(s) = 1 | − 2·2-s + 4-s − 2·5-s + 4·10-s − 4·11-s − 4·13-s + 16-s − 4·17-s − 2·20-s + 8·22-s + 2·23-s + 3·25-s + 8·26-s + 2·29-s − 12·31-s + 2·32-s + 8·34-s + 10·41-s + 10·43-s − 4·44-s − 4·46-s − 4·47-s − 6·50-s − 4·52-s + 8·53-s + 8·55-s − 4·58-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s − 0.894·5-s + 1.26·10-s − 1.20·11-s − 1.10·13-s + 1/4·16-s − 0.970·17-s − 0.447·20-s + 1.70·22-s + 0.417·23-s + 3/5·25-s + 1.56·26-s + 0.371·29-s − 2.15·31-s + 0.353·32-s + 1.37·34-s + 1.56·41-s + 1.52·43-s − 0.603·44-s − 0.589·46-s − 0.583·47-s − 0.848·50-s − 0.554·52-s + 1.09·53-s + 1.07·55-s − 0.525·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4954119139\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4954119139\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 45 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 10 T + 99 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 10 T + 109 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 114 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 59 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 22 T + 253 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T + 142 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 24 T + 294 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 155 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 12 T + 198 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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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
−9.203744175533908713375518911221, −8.997027504158089921985324218281, −8.348487189255984960524026739309, −8.225066809539496424417292086703, −7.72283621249452497011192608183, −7.57641077880455419955925989544, −7.14066522888487143731143403883, −6.70766403426222498241964632193, −6.33794296564851945076478752541, −5.59872840912335525457093924731, −5.13419491357220956258016225759, −5.10953837743043656398701519902, −4.19079668751028886155210833294, −4.11546797065778414619698017820, −3.41477777042949311280475784776, −2.78205283994588028028733380699, −2.37476320991162787116112382140, −1.92836876658929108662529847577, −0.62219726315800900212348935724, −0.56406900120573674418013856149,
0.56406900120573674418013856149, 0.62219726315800900212348935724, 1.92836876658929108662529847577, 2.37476320991162787116112382140, 2.78205283994588028028733380699, 3.41477777042949311280475784776, 4.11546797065778414619698017820, 4.19079668751028886155210833294, 5.10953837743043656398701519902, 5.13419491357220956258016225759, 5.59872840912335525457093924731, 6.33794296564851945076478752541, 6.70766403426222498241964632193, 7.14066522888487143731143403883, 7.57641077880455419955925989544, 7.72283621249452497011192608183, 8.225066809539496424417292086703, 8.348487189255984960524026739309, 8.997027504158089921985324218281, 9.203744175533908713375518911221
