| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 7-s + 8-s + 9-s − 2·11-s − 12-s − 14-s + 16-s − 17-s + 18-s − 4·19-s + 21-s − 2·22-s − 24-s − 27-s − 28-s + 32-s + 2·33-s − 34-s + 36-s + 10·37-s − 4·38-s + 8·41-s + 42-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.603·11-s − 0.288·12-s − 0.267·14-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.917·19-s + 0.218·21-s − 0.426·22-s − 0.204·24-s − 0.192·27-s − 0.188·28-s + 0.176·32-s + 0.348·33-s − 0.171·34-s + 1/6·36-s + 1.64·37-s − 0.648·38-s + 1.24·41-s + 0.154·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 + T \) | |
| good | 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−15.99448331639620, −15.58565287975147, −14.94572740148991, −14.57233459646787, −13.70215729948921, −13.30153451955082, −12.78283904065730, −12.32882371661836, −11.75822299698362, −10.98048528666116, −10.78071852296066, −10.06126437606712, −9.449312536883519, −8.730567314903203, −7.947147702516100, −7.391295961546993, −6.739714771430784, −6.016137095820315, −5.793353296735066, −4.832386277428790, −4.394348115795541, −3.697609746648205, −2.767785637524733, −2.224350462314564, −1.105102355157628, 0,
1.105102355157628, 2.224350462314564, 2.767785637524733, 3.697609746648205, 4.394348115795541, 4.832386277428790, 5.793353296735066, 6.016137095820315, 6.739714771430784, 7.391295961546993, 7.947147702516100, 8.730567314903203, 9.449312536883519, 10.06126437606712, 10.78071852296066, 10.98048528666116, 11.75822299698362, 12.32882371661836, 12.78283904065730, 13.30153451955082, 13.70215729948921, 14.57233459646787, 14.94572740148991, 15.58565287975147, 15.99448331639620
