| L(s) = 1 | + 7-s − 3·9-s − 11-s − 13-s + 4·17-s + 5·19-s − 8·23-s − 5·25-s − 29-s + 3·31-s + 8·37-s + 9·41-s − 5·43-s + 4·47-s − 6·49-s − 8·53-s − 11·59-s − 2·61-s − 3·63-s − 3·67-s + 5·71-s − 14·73-s − 77-s − 3·79-s + 9·81-s − 4·83-s + 11·89-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 9-s − 0.301·11-s − 0.277·13-s + 0.970·17-s + 1.14·19-s − 1.66·23-s − 25-s − 0.185·29-s + 0.538·31-s + 1.31·37-s + 1.40·41-s − 0.762·43-s + 0.583·47-s − 6/7·49-s − 1.09·53-s − 1.43·59-s − 0.256·61-s − 0.377·63-s − 0.366·67-s + 0.593·71-s − 1.63·73-s − 0.113·77-s − 0.337·79-s + 81-s − 0.439·83-s + 1.16·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 619 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 + 11 T + p T^{2} \) | 1.59.l |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 3 T + p T^{2} \) | 1.79.d |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 11 T + p T^{2} \) | 1.89.al |
| 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
−7.84096053102356722902809834368, −7.50533583251773384277558021331, −6.14553857202612444331339270070, −5.83683847547981975791950316556, −5.02705669628938247207984410548, −4.15862810004211349373851107138, −3.21812128830068536763702812032, −2.48873487190086254787358137136, −1.37300443327958134067349033233, 0,
1.37300443327958134067349033233, 2.48873487190086254787358137136, 3.21812128830068536763702812032, 4.15862810004211349373851107138, 5.02705669628938247207984410548, 5.83683847547981975791950316556, 6.14553857202612444331339270070, 7.50533583251773384277558021331, 7.84096053102356722902809834368
