| L(s) = 1 | + 4·7-s − 3·9-s + 4·11-s + 13-s + 2·17-s − 4·19-s − 2·23-s − 5·25-s + 2·29-s + 8·31-s − 2·37-s − 12·41-s + 6·43-s − 8·47-s + 9·49-s − 10·53-s + 10·61-s − 12·63-s − 4·67-s + 8·71-s + 6·73-s + 16·77-s + 6·79-s + 9·81-s − 18·89-s + 4·91-s + 2·97-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 9-s + 1.20·11-s + 0.277·13-s + 0.485·17-s − 0.917·19-s − 0.417·23-s − 25-s + 0.371·29-s + 1.43·31-s − 0.328·37-s − 1.87·41-s + 0.914·43-s − 1.16·47-s + 9/7·49-s − 1.37·53-s + 1.28·61-s − 1.51·63-s − 0.488·67-s + 0.949·71-s + 0.702·73-s + 1.82·77-s + 0.675·79-s + 81-s − 1.90·89-s + 0.419·91-s + 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| 937 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 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
−14.07933702087265, −13.86130119450646, −13.10397939081309, −12.34874053332705, −11.95239448889105, −11.55673976010208, −11.24767685260756, −10.71178694081655, −10.06304552605430, −9.606509846512492, −8.885740442932645, −8.472549143004897, −8.047614584438273, −7.823489380140911, −6.704036796842054, −6.546703760254346, −5.852954161164558, −5.277084049817362, −4.784410524024901, −4.155460933816637, −3.689190547680280, −2.945759109640480, −2.163527179065668, −1.636700131264540, −1.028923684677001, 0,
1.028923684677001, 1.636700131264540, 2.163527179065668, 2.945759109640480, 3.689190547680280, 4.155460933816637, 4.784410524024901, 5.277084049817362, 5.852954161164558, 6.546703760254346, 6.704036796842054, 7.823489380140911, 8.047614584438273, 8.472549143004897, 8.885740442932645, 9.606509846512492, 10.06304552605430, 10.71178694081655, 11.24767685260756, 11.55673976010208, 11.95239448889105, 12.34874053332705, 13.10397939081309, 13.86130119450646, 14.07933702087265
