| L(s) = 1 | + 2-s + 4-s + 4·7-s + 8-s − 3·9-s + 2·11-s + 2·13-s + 4·14-s + 16-s + 2·17-s − 3·18-s − 2·19-s + 2·22-s − 23-s + 2·26-s + 4·28-s + 2·29-s + 32-s + 2·34-s − 3·36-s + 4·37-s − 2·38-s + 6·41-s − 10·43-s + 2·44-s − 46-s + 9·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.51·7-s + 0.353·8-s − 9-s + 0.603·11-s + 0.554·13-s + 1.06·14-s + 1/4·16-s + 0.485·17-s − 0.707·18-s − 0.458·19-s + 0.426·22-s − 0.208·23-s + 0.392·26-s + 0.755·28-s + 0.371·29-s + 0.176·32-s + 0.342·34-s − 1/2·36-s + 0.657·37-s − 0.324·38-s + 0.937·41-s − 1.52·43-s + 0.301·44-s − 0.147·46-s + 9/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.955653039\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.955653039\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−9.928280836928168979628777920327, −8.651590213691851109739649270908, −8.276543750439528714060318536349, −7.29468472679713756407763618174, −6.21475081018625957585151869260, −5.50098265982593501879266203197, −4.63556271802396925268184937965, −3.74894127876684642815312676386, −2.53491101319297313540699620659, −1.35564478109443112808861515260,
1.35564478109443112808861515260, 2.53491101319297313540699620659, 3.74894127876684642815312676386, 4.63556271802396925268184937965, 5.50098265982593501879266203197, 6.21475081018625957585151869260, 7.29468472679713756407763618174, 8.276543750439528714060318536349, 8.651590213691851109739649270908, 9.928280836928168979628777920327
