| L(s) = 1 | + 3-s + 2·5-s − 2·7-s − 2·9-s + 6·11-s − 13-s + 2·15-s + 6·17-s + 4·19-s − 2·21-s − 25-s − 5·27-s + 9·29-s − 3·31-s + 6·33-s − 4·35-s − 2·37-s − 39-s − 41-s − 4·45-s + 47-s − 3·49-s + 6·51-s − 12·53-s + 12·55-s + 4·57-s + 12·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s − 0.755·7-s − 2/3·9-s + 1.80·11-s − 0.277·13-s + 0.516·15-s + 1.45·17-s + 0.917·19-s − 0.436·21-s − 1/5·25-s − 0.962·27-s + 1.67·29-s − 0.538·31-s + 1.04·33-s − 0.676·35-s − 0.328·37-s − 0.160·39-s − 0.156·41-s − 0.596·45-s + 0.145·47-s − 3/7·49-s + 0.840·51-s − 1.64·53-s + 1.61·55-s + 0.529·57-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.229211702\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.229211702\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + T + p T^{2} \) | 1.41.b |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 - 13 T + p T^{2} \) | 1.71.an |
| 73 | \( 1 - 5 T + p T^{2} \) | 1.73.af |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−7.965548164658750417067266539195, −6.83518001736639639005107260689, −6.57350995225394492849448624435, −5.67988377062141005215323303820, −5.25212318350299275174801514220, −3.99613470939665463204813407676, −3.39116045392570759391414671647, −2.77701526665756426549673154334, −1.77311977753816525953753848355, −0.892804709800706597556236172487,
0.892804709800706597556236172487, 1.77311977753816525953753848355, 2.77701526665756426549673154334, 3.39116045392570759391414671647, 3.99613470939665463204813407676, 5.25212318350299275174801514220, 5.67988377062141005215323303820, 6.57350995225394492849448624435, 6.83518001736639639005107260689, 7.965548164658750417067266539195
