| L(s) = 1 | + 2·2-s + 3·4-s − 2·7-s + 4·8-s + 9-s + 2·11-s − 4·14-s + 5·16-s + 2·18-s + 4·22-s − 12·23-s + 6·25-s − 6·28-s + 20·29-s + 6·32-s + 3·36-s − 4·37-s + 8·43-s + 6·44-s − 24·46-s − 3·49-s + 12·50-s + 8·53-s − 8·56-s + 40·58-s − 2·63-s + 7·64-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.755·7-s + 1.41·8-s + 1/3·9-s + 0.603·11-s − 1.06·14-s + 5/4·16-s + 0.471·18-s + 0.852·22-s − 2.50·23-s + 6/5·25-s − 1.13·28-s + 3.71·29-s + 1.06·32-s + 1/2·36-s − 0.657·37-s + 1.21·43-s + 0.904·44-s − 3.53·46-s − 3/7·49-s + 1.69·50-s + 1.09·53-s − 1.06·56-s + 5.25·58-s − 0.251·63-s + 7/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.234907127\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.234907127\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.822664775845247131815531688861, −8.734399327056068821213499579504, −7.74249612429138712743707138359, −7.72339303053252541131209387915, −6.69373326676767816887250497086, −6.46917181705411800265373292912, −6.35062410158064670794655966913, −5.56849277330524351488966193226, −5.01244554431640501263959251141, −4.25842178547984493307347569445, −4.22775784486927931972933466181, −3.34833436263269527102959669443, −2.84811781912171505509456779453, −2.18727745386797108535245912439, −1.12176628667720465571658579928,
1.12176628667720465571658579928, 2.18727745386797108535245912439, 2.84811781912171505509456779453, 3.34833436263269527102959669443, 4.22775784486927931972933466181, 4.25842178547984493307347569445, 5.01244554431640501263959251141, 5.56849277330524351488966193226, 6.35062410158064670794655966913, 6.46917181705411800265373292912, 6.69373326676767816887250497086, 7.72339303053252541131209387915, 7.74249612429138712743707138359, 8.734399327056068821213499579504, 8.822664775845247131815531688861
