| L(s) = 1 | + 2·2-s + 2·4-s − 7-s + 6·11-s − 7·13-s − 2·14-s − 4·16-s + 4·17-s + 5·19-s + 12·22-s + 2·23-s − 14·26-s − 2·28-s + 8·29-s + 3·31-s − 8·32-s + 8·34-s + 37-s + 10·38-s − 2·41-s − 11·43-s + 12·44-s + 4·46-s + 4·47-s − 6·49-s − 14·52-s + 16·58-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 0.377·7-s + 1.80·11-s − 1.94·13-s − 0.534·14-s − 16-s + 0.970·17-s + 1.14·19-s + 2.55·22-s + 0.417·23-s − 2.74·26-s − 0.377·28-s + 1.48·29-s + 0.538·31-s − 1.41·32-s + 1.37·34-s + 0.164·37-s + 1.62·38-s − 0.312·41-s − 1.67·43-s + 1.80·44-s + 0.589·46-s + 0.583·47-s − 6/7·49-s − 1.94·52-s + 2.10·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.483723885\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.483723885\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 37 | \( 1 - T \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 + 7 T + p T^{2} \) | 1.13.h |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 11 T + p T^{2} \) | 1.97.l |
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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
−7.44060526077355563557869029209, −6.87957477493459379473512770312, −6.37788529251015560317933473874, −5.57782711337049296818326691106, −4.84437392348548814728309839130, −4.46791638101019170184784495260, −3.36512218578266685821806080208, −3.14120709355627988971745537233, −2.06000754288520141546374566152, −0.845245944304518618775833635646,
0.845245944304518618775833635646, 2.06000754288520141546374566152, 3.14120709355627988971745537233, 3.36512218578266685821806080208, 4.46791638101019170184784495260, 4.84437392348548814728309839130, 5.57782711337049296818326691106, 6.37788529251015560317933473874, 6.87957477493459379473512770312, 7.44060526077355563557869029209
