| L(s) = 1 | + 3·3-s − 5-s + 6·9-s − 11-s + 2·13-s − 3·15-s + 3·17-s + 5·19-s − 3·23-s − 4·25-s + 9·27-s − 6·29-s − 31-s − 3·33-s − 5·37-s + 6·39-s − 10·41-s − 4·43-s − 6·45-s + 47-s + 9·51-s − 9·53-s + 55-s + 15·57-s + 3·59-s + 3·61-s − 2·65-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.447·5-s + 2·9-s − 0.301·11-s + 0.554·13-s − 0.774·15-s + 0.727·17-s + 1.14·19-s − 0.625·23-s − 4/5·25-s + 1.73·27-s − 1.11·29-s − 0.179·31-s − 0.522·33-s − 0.821·37-s + 0.960·39-s − 1.56·41-s − 0.609·43-s − 0.894·45-s + 0.145·47-s + 1.26·51-s − 1.23·53-s + 0.134·55-s + 1.98·57-s + 0.390·59-s + 0.384·61-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.221527878\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.221527878\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - 3 T + p T^{2} \) | 1.61.ad |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−11.32363161411022171833837316965, −10.03255565958818673784718261940, −9.454419896289841119816933895926, −8.351257157353401744523838901572, −7.88711144460196356605409250981, −6.97371055422473493569992163867, −5.37275328661669194907215278623, −3.85088303636964375675997178618, −3.24705941901609595501905536901, −1.80924568747514375985943654352,
1.80924568747514375985943654352, 3.24705941901609595501905536901, 3.85088303636964375675997178618, 5.37275328661669194907215278623, 6.97371055422473493569992163867, 7.88711144460196356605409250981, 8.351257157353401744523838901572, 9.454419896289841119816933895926, 10.03255565958818673784718261940, 11.32363161411022171833837316965
