| L(s) = 1 | − 2·7-s + 11-s + 4·13-s + 6·17-s − 4·19-s + 6·23-s − 5·25-s + 6·29-s − 8·31-s + 10·37-s − 6·41-s + 8·43-s − 6·47-s − 3·49-s − 8·61-s − 4·67-s + 6·71-s + 2·73-s − 2·77-s − 14·79-s + 12·83-s + 6·89-s − 8·91-s + 14·97-s + 6·101-s + 4·103-s + 12·107-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 0.301·11-s + 1.10·13-s + 1.45·17-s − 0.917·19-s + 1.25·23-s − 25-s + 1.11·29-s − 1.43·31-s + 1.64·37-s − 0.937·41-s + 1.21·43-s − 0.875·47-s − 3/7·49-s − 1.02·61-s − 0.488·67-s + 0.712·71-s + 0.234·73-s − 0.227·77-s − 1.57·79-s + 1.31·83-s + 0.635·89-s − 0.838·91-s + 1.42·97-s + 0.597·101-s + 0.394·103-s + 1.16·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.952467348\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.952467348\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−7.990007746064461326872519078273, −7.38377174658580997166042366079, −6.40864434101561615097433391410, −6.11485173072225725039951775848, −5.26285470699018847158598494284, −4.30603307745423902511409530972, −3.52394436125708712603168440916, −2.97366741962795049276596787100, −1.74289628806760359792201029535, −0.74841655188926250687663175586,
0.74841655188926250687663175586, 1.74289628806760359792201029535, 2.97366741962795049276596787100, 3.52394436125708712603168440916, 4.30603307745423902511409530972, 5.26285470699018847158598494284, 6.11485173072225725039951775848, 6.40864434101561615097433391410, 7.38377174658580997166042366079, 7.990007746064461326872519078273
