| L(s) = 1 | + 2·2-s − 3·3-s − 5·5-s − 6·6-s + 7·7-s − 8·8-s − 10·10-s + 32·11-s + 30·13-s + 14·14-s + 15·15-s − 16·16-s + 70·17-s − 15·19-s − 21·21-s + 64·22-s + 42·23-s + 24·24-s + 60·26-s + 27·27-s + 180·29-s + 30·30-s + 85·31-s − 96·33-s + 140·34-s − 35·35-s − 113·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 0.447·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s − 0.316·10-s + 0.877·11-s + 0.640·13-s + 0.267·14-s + 0.258·15-s − 1/4·16-s + 0.998·17-s − 0.181·19-s − 0.218·21-s + 0.620·22-s + 0.380·23-s + 0.204·24-s + 0.452·26-s + 0.192·27-s + 1.15·29-s + 0.182·30-s + 0.492·31-s − 0.506·33-s + 0.706·34-s − 0.169·35-s − 0.502·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.694921008\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.694921008\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 3 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 5 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 7 | $C_2$ | \( 1 - p T + p^{3} T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 - 32 T - 307 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 15 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 70 T - 13 T^{2} - 70 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 15 T - 6634 T^{2} + 15 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 42 T - 10403 T^{2} - 42 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 90 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 85 T - 22566 T^{2} - 85 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 113 T - 37884 T^{2} + 113 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 4 p T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 169 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 326 T + 2453 T^{2} + 326 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 44 T - 146941 T^{2} - 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 782 T + 406145 T^{2} - 782 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 658 T + 205983 T^{2} + 658 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 1071 T + 846278 T^{2} + 1071 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 344 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 431 T - 203256 T^{2} + 431 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 397 T - 335430 T^{2} + 397 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 680 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 1534 T + 1648187 T^{2} + 1534 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 1234 T + p^{3} T^{2} )^{2} \) |
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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
−12.19110391706131700324921014597, −11.79605409899090170531555753231, −11.16947351071085957933188993392, −11.05610597073713264910178020163, −10.25858567311493747497484406325, −9.867171761063229073963458711247, −9.056532130990221418521825129986, −8.806607706237343609875082652829, −7.979143513242925274148446946439, −7.76476334497052701477306377107, −6.68202772492317043514094277429, −6.62072723250125388669360973902, −5.68468458082407195861167936968, −5.48464598310874231782762791282, −4.42295643709386469415967896040, −4.36145119301133033316654203511, −3.42769957289789071639614529639, −2.83609917034919149304416306544, −1.50974537019828038947392195338, −0.70651018429537439013075414551,
0.70651018429537439013075414551, 1.50974537019828038947392195338, 2.83609917034919149304416306544, 3.42769957289789071639614529639, 4.36145119301133033316654203511, 4.42295643709386469415967896040, 5.48464598310874231782762791282, 5.68468458082407195861167936968, 6.62072723250125388669360973902, 6.68202772492317043514094277429, 7.76476334497052701477306377107, 7.979143513242925274148446946439, 8.806607706237343609875082652829, 9.056532130990221418521825129986, 9.867171761063229073963458711247, 10.25858567311493747497484406325, 11.05610597073713264910178020163, 11.16947351071085957933188993392, 11.79605409899090170531555753231, 12.19110391706131700324921014597
