| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 12-s − 2·13-s + 16-s − 18-s − 8·23-s − 24-s − 5·25-s + 2·26-s + 27-s − 4·29-s − 32-s + 36-s + 4·37-s − 2·39-s + 8·43-s + 8·46-s − 12·47-s + 48-s − 7·49-s + 5·50-s − 2·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.288·12-s − 0.554·13-s + 1/4·16-s − 0.235·18-s − 1.66·23-s − 0.204·24-s − 25-s + 0.392·26-s + 0.192·27-s − 0.742·29-s − 0.176·32-s + 1/6·36-s + 0.657·37-s − 0.320·39-s + 1.21·43-s + 1.17·46-s − 1.75·47-s + 0.144·48-s − 49-s + 0.707·50-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209814 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209814 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 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.k |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−13.12376499042288, −12.82069574450362, −12.29683213057936, −11.73397197836956, −11.41206795205425, −10.86061283422940, −10.27945619423845, −9.814541432398813, −9.493476371477883, −9.204919376393831, −8.285334769750139, −8.164870880749679, −7.639967571871631, −7.310656694312770, −6.390024844052945, −6.348333052041785, −5.530552870883062, −5.037092349220749, −4.240834640952207, −3.881776682845227, −3.223577549399149, −2.632529858410545, −1.915921517261931, −1.767470940324653, −0.7024902064940092, 0,
0.7024902064940092, 1.767470940324653, 1.915921517261931, 2.632529858410545, 3.223577549399149, 3.881776682845227, 4.240834640952207, 5.037092349220749, 5.530552870883062, 6.348333052041785, 6.390024844052945, 7.310656694312770, 7.639967571871631, 8.164870880749679, 8.285334769750139, 9.204919376393831, 9.493476371477883, 9.814541432398813, 10.27945619423845, 10.86061283422940, 11.41206795205425, 11.73397197836956, 12.29683213057936, 12.82069574450362, 13.12376499042288
