| L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s − 7-s + 8-s + 9-s + 6·11-s + 2·12-s − 4·13-s − 14-s + 16-s + 3·17-s + 18-s + 4·19-s − 2·21-s + 6·22-s + 9·23-s + 2·24-s − 5·25-s − 4·26-s − 4·27-s − 28-s − 5·31-s + 32-s + 12·33-s + 3·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 1.80·11-s + 0.577·12-s − 1.10·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s + 0.235·18-s + 0.917·19-s − 0.436·21-s + 1.27·22-s + 1.87·23-s + 0.408·24-s − 25-s − 0.784·26-s − 0.769·27-s − 0.188·28-s − 0.898·31-s + 0.176·32-s + 2.08·33-s + 0.514·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.106204617\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.106204617\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 29 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−9.456458443533822732896897031223, −8.677117938950219848302186352545, −7.50723804849512570961136065812, −7.16366469160968703287751003216, −6.09283190358559230018773703443, −5.18599408584181814805220584333, −4.07172783854566247841696623218, −3.38601028447971707410746827239, −2.63516822507432546692486958838, −1.39345694576822601394248550654,
1.39345694576822601394248550654, 2.63516822507432546692486958838, 3.38601028447971707410746827239, 4.07172783854566247841696623218, 5.18599408584181814805220584333, 6.09283190358559230018773703443, 7.16366469160968703287751003216, 7.50723804849512570961136065812, 8.677117938950219848302186352545, 9.456458443533822732896897031223
