| L(s) = 1 | + 4·11-s + 13-s − 6·17-s + 4·19-s + 2·29-s − 4·31-s + 6·37-s + 6·41-s − 8·43-s − 7·49-s + 2·53-s − 4·59-s − 10·61-s − 12·67-s + 4·71-s − 14·73-s − 16·79-s + 12·83-s − 2·89-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.20·11-s + 0.277·13-s − 1.45·17-s + 0.917·19-s + 0.371·29-s − 0.718·31-s + 0.986·37-s + 0.937·41-s − 1.21·43-s − 49-s + 0.274·53-s − 0.520·59-s − 1.28·61-s − 1.46·67-s + 0.474·71-s − 1.63·73-s − 1.80·79-s + 1.31·83-s − 0.211·89-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.82970308814078, −15.03329197583722, −14.75170238202599, −14.09596229650379, −13.54462618269859, −13.13587569079189, −12.47501120893156, −11.77116003673693, −11.47767034652887, −10.89673660400005, −10.28184867995865, −9.471900079540558, −9.177038784973002, −8.638624174375151, −7.901250552112171, −7.269676281303710, −6.650984352013632, −6.182003938740287, −5.535233703383217, −4.578988382754620, −4.285652804857263, −3.417601181347216, −2.789020899216971, −1.799984620569868, −1.180737568371585, 0,
1.180737568371585, 1.799984620569868, 2.789020899216971, 3.417601181347216, 4.285652804857263, 4.578988382754620, 5.535233703383217, 6.182003938740287, 6.650984352013632, 7.269676281303710, 7.901250552112171, 8.638624174375151, 9.177038784973002, 9.471900079540558, 10.28184867995865, 10.89673660400005, 11.47767034652887, 11.77116003673693, 12.47501120893156, 13.13587569079189, 13.54462618269859, 14.09596229650379, 14.75170238202599, 15.03329197583722, 15.82970308814078
