| L(s) = 1 | − 3-s − 5-s + 9-s − 3·11-s − 13-s + 15-s + 6·17-s − 8·19-s + 6·23-s − 4·25-s − 27-s − 5·29-s − 9·31-s + 3·33-s − 8·37-s + 39-s − 8·41-s − 8·43-s − 45-s + 12·47-s − 6·51-s − 5·53-s + 3·55-s + 8·57-s − 15·59-s + 65-s − 4·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.904·11-s − 0.277·13-s + 0.258·15-s + 1.45·17-s − 1.83·19-s + 1.25·23-s − 4/5·25-s − 0.192·27-s − 0.928·29-s − 1.61·31-s + 0.522·33-s − 1.31·37-s + 0.160·39-s − 1.24·41-s − 1.21·43-s − 0.149·45-s + 1.75·47-s − 0.840·51-s − 0.686·53-s + 0.404·55-s + 1.05·57-s − 1.95·59-s + 0.124·65-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 + 9 T + p T^{2} \) | 1.31.j |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 5 T + p T^{2} \) | 1.53.f |
| 59 | \( 1 + 15 T + p T^{2} \) | 1.59.p |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 - 13 T + p T^{2} \) | 1.83.an |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−14.93074879398435, −14.39550784582006, −13.69499812361545, −13.13715431032120, −12.75443850435295, −12.19674570475384, −11.93490370656780, −11.11724899086348, −10.62953228486303, −10.53831069302671, −9.725245618795045, −9.179049563747944, −8.595283306208689, −7.927563887007457, −7.567352844881929, −7.003766894446504, −6.432464697730368, −5.704395266353393, −5.231300932939133, −4.879993919396061, −3.929644834119661, −3.586459533285297, −2.807439910526381, −1.962443601352949, −1.369147248761617, 0, 0,
1.369147248761617, 1.962443601352949, 2.807439910526381, 3.586459533285297, 3.929644834119661, 4.879993919396061, 5.231300932939133, 5.704395266353393, 6.432464697730368, 7.003766894446504, 7.567352844881929, 7.927563887007457, 8.595283306208689, 9.179049563747944, 9.725245618795045, 10.53831069302671, 10.62953228486303, 11.11724899086348, 11.93490370656780, 12.19674570475384, 12.75443850435295, 13.13715431032120, 13.69499812361545, 14.39550784582006, 14.93074879398435
