| L(s) = 1 | − 2-s − 4-s + 3·8-s − 11-s − 2·13-s − 16-s + 6·17-s + 4·19-s + 22-s + 8·23-s + 2·26-s − 6·29-s − 8·31-s − 5·32-s − 6·34-s − 6·37-s − 4·38-s − 6·41-s + 4·43-s + 44-s − 8·46-s + 8·47-s + 2·52-s − 10·53-s + 6·58-s − 12·59-s + 10·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s − 0.301·11-s − 0.554·13-s − 1/4·16-s + 1.45·17-s + 0.917·19-s + 0.213·22-s + 1.66·23-s + 0.392·26-s − 1.11·29-s − 1.43·31-s − 0.883·32-s − 1.02·34-s − 0.986·37-s − 0.648·38-s − 0.937·41-s + 0.609·43-s + 0.150·44-s − 1.17·46-s + 1.16·47-s + 0.277·52-s − 1.37·53-s + 0.787·58-s − 1.56·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−13.93599919771586, −13.19411209653544, −12.87687847143530, −12.39415639073395, −11.90674703794434, −11.18685963589435, −10.78491956541902, −10.38830439965085, −9.724077614788567, −9.358342693860807, −9.116611872702427, −8.407336013337233, −7.860073956379809, −7.379978639467404, −7.221732955244206, −6.399379152495483, −5.445179470768899, −5.261710249133261, −4.921953361289190, −3.929966207024007, −3.516258341730089, −2.963082354163855, −2.065297308809048, −1.398197217414900, −0.8038524817253572, 0,
0.8038524817253572, 1.398197217414900, 2.065297308809048, 2.963082354163855, 3.516258341730089, 3.929966207024007, 4.921953361289190, 5.261710249133261, 5.445179470768899, 6.399379152495483, 7.221732955244206, 7.379978639467404, 7.860073956379809, 8.407336013337233, 9.116611872702427, 9.358342693860807, 9.724077614788567, 10.38830439965085, 10.78491956541902, 11.18685963589435, 11.90674703794434, 12.39415639073395, 12.87687847143530, 13.19411209653544, 13.93599919771586
