| L(s) = 1 | − 2-s − 3·3-s + 4-s + 3·6-s − 2·7-s − 8-s + 6·9-s + 2·11-s − 3·12-s + 2·14-s + 16-s + 2·17-s − 6·18-s − 2·19-s + 6·21-s − 2·22-s + 23-s + 3·24-s − 9·27-s − 2·28-s − 10·29-s − 3·31-s − 32-s − 6·33-s − 2·34-s + 6·36-s + 8·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s + 1/2·4-s + 1.22·6-s − 0.755·7-s − 0.353·8-s + 2·9-s + 0.603·11-s − 0.866·12-s + 0.534·14-s + 1/4·16-s + 0.485·17-s − 1.41·18-s − 0.458·19-s + 1.30·21-s − 0.426·22-s + 0.208·23-s + 0.612·24-s − 1.73·27-s − 0.377·28-s − 1.85·29-s − 0.538·31-s − 0.176·32-s − 1.04·33-s − 0.342·34-s + 36-s + 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 18 T + p T^{2} \) | 1.83.s |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−13.13326770626117, −12.87082199753025, −12.63527014640523, −12.03337952468253, −11.52296898965457, −11.12612070491296, −10.93056026075572, −10.31046881030765, −9.717570090448096, −9.530712767367727, −9.043005601920243, −8.327426492077911, −7.552226271572031, −7.363420383800422, −6.737622306317213, −6.265657458750705, −5.891876544763991, −5.569122141377719, −4.829168560811604, −4.236431388793556, −3.753125083157585, −3.044859869107522, −2.274276243329845, −1.426334611247503, −1.095100153294620, 0, 0,
1.095100153294620, 1.426334611247503, 2.274276243329845, 3.044859869107522, 3.753125083157585, 4.236431388793556, 4.829168560811604, 5.569122141377719, 5.891876544763991, 6.265657458750705, 6.737622306317213, 7.363420383800422, 7.552226271572031, 8.327426492077911, 9.043005601920243, 9.530712767367727, 9.717570090448096, 10.31046881030765, 10.93056026075572, 11.12612070491296, 11.52296898965457, 12.03337952468253, 12.63527014640523, 12.87082199753025, 13.13326770626117
