| L(s) = 1 | − 3-s + 2·5-s − 7-s + 9-s + 6·11-s − 2·13-s − 2·15-s − 8·17-s + 19-s + 21-s − 8·23-s − 25-s − 27-s − 2·29-s − 6·31-s − 6·33-s − 2·35-s − 4·37-s + 2·39-s − 2·41-s + 4·43-s + 2·45-s − 6·47-s + 49-s + 8·51-s + 6·53-s + 12·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s + 1.80·11-s − 0.554·13-s − 0.516·15-s − 1.94·17-s + 0.229·19-s + 0.218·21-s − 1.66·23-s − 1/5·25-s − 0.192·27-s − 0.371·29-s − 1.07·31-s − 1.04·33-s − 0.338·35-s − 0.657·37-s + 0.320·39-s − 0.312·41-s + 0.609·43-s + 0.298·45-s − 0.875·47-s + 1/7·49-s + 1.12·51-s + 0.824·53-s + 1.61·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 19 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 8 T + p T^{2} \) | 1.17.i |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.496969463179047971373586914716, −7.23650727576498476574632538463, −6.66397915649223030008175795827, −6.11097501127765743273286851064, −5.40977045329054431700195589380, −4.31659367087893074191268677199, −3.77329238040444330032205728695, −2.27661561577792034142292009878, −1.60169466870749720290504692333, 0,
1.60169466870749720290504692333, 2.27661561577792034142292009878, 3.77329238040444330032205728695, 4.31659367087893074191268677199, 5.40977045329054431700195589380, 6.11097501127765743273286851064, 6.66397915649223030008175795827, 7.23650727576498476574632538463, 8.496969463179047971373586914716
