| L(s) = 1 | − 2-s + 4-s − 8-s − 3·9-s + 6·11-s + 13-s + 16-s + 17-s + 3·18-s − 6·22-s − 3·23-s − 26-s − 7·31-s − 32-s − 34-s − 3·36-s + 2·37-s + 5·41-s − 6·43-s + 6·44-s + 3·46-s − 47-s + 52-s + 8·59-s − 6·61-s + 7·62-s + 64-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 9-s + 1.80·11-s + 0.277·13-s + 1/4·16-s + 0.242·17-s + 0.707·18-s − 1.27·22-s − 0.625·23-s − 0.196·26-s − 1.25·31-s − 0.176·32-s − 0.171·34-s − 1/2·36-s + 0.328·37-s + 0.780·41-s − 0.914·43-s + 0.904·44-s + 0.442·46-s − 0.145·47-s + 0.138·52-s + 1.04·59-s − 0.768·61-s + 0.889·62-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 - 3 T + p T^{2} \) | 1.97.ad |
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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
−15.33109954646609, −14.64416089146900, −14.42556506164617, −13.95896278929599, −13.20281102818848, −12.53320491411900, −11.96000223622381, −11.51256387731598, −11.19104038017159, −10.52331010554650, −9.797007398178876, −9.378355083289823, −8.791510568763517, −8.494901191612089, −7.745840801259970, −7.155562781317419, −6.502645524992681, −6.028031531076711, −5.529421773794436, −4.601978940599400, −3.784154974709013, −3.394513746025475, −2.473533264345100, −1.711858627704317, −1.014139113920021, 0,
1.014139113920021, 1.711858627704317, 2.473533264345100, 3.394513746025475, 3.784154974709013, 4.601978940599400, 5.529421773794436, 6.028031531076711, 6.502645524992681, 7.155562781317419, 7.745840801259970, 8.494901191612089, 8.791510568763517, 9.378355083289823, 9.797007398178876, 10.52331010554650, 11.19104038017159, 11.51256387731598, 11.96000223622381, 12.53320491411900, 13.20281102818848, 13.95896278929599, 14.42556506164617, 14.64416089146900, 15.33109954646609
