| L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s − 2·11-s + 5·13-s + 14-s + 16-s − 3·17-s − 20-s + 2·22-s − 23-s − 4·25-s − 5·26-s − 28-s − 2·29-s − 4·31-s − 32-s + 3·34-s + 35-s + 8·37-s + 40-s + 6·41-s + 7·43-s − 2·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s − 0.353·8-s + 0.316·10-s − 0.603·11-s + 1.38·13-s + 0.267·14-s + 1/4·16-s − 0.727·17-s − 0.223·20-s + 0.426·22-s − 0.208·23-s − 4/5·25-s − 0.980·26-s − 0.188·28-s − 0.371·29-s − 0.718·31-s − 0.176·32-s + 0.514·34-s + 0.169·35-s + 1.31·37-s + 0.158·40-s + 0.937·41-s + 1.06·43-s − 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 149454 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149454 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.552007822\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.552007822\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 19 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 13 T + p T^{2} \) | 1.71.an |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 3 T + p T^{2} \) | 1.79.ad |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
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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
−13.23922590835932, −12.85911119715165, −12.46225099318719, −11.62101583034374, −11.46448360253657, −10.84036228228754, −10.60322670962349, −9.995710430641757, −9.301383666202031, −9.121505363014880, −8.488612520928977, −8.038958010112482, −7.524538154027213, −7.199501214160300, −6.389863264046832, −6.027934222310171, −5.638961567926730, −4.824537849987044, −4.070089626707778, −3.770074031216977, −3.122550287472994, −2.310498308672154, −1.999959485854060, −0.9147461080932249, −0.5185007086826063,
0.5185007086826063, 0.9147461080932249, 1.999959485854060, 2.310498308672154, 3.122550287472994, 3.770074031216977, 4.070089626707778, 4.824537849987044, 5.638961567926730, 6.027934222310171, 6.389863264046832, 7.199501214160300, 7.524538154027213, 8.038958010112482, 8.488612520928977, 9.121505363014880, 9.301383666202031, 9.995710430641757, 10.60322670962349, 10.84036228228754, 11.46448360253657, 11.62101583034374, 12.46225099318719, 12.85911119715165, 13.23922590835932
