| L(s) = 1 | + 2·3-s − 3·5-s − 7-s + 9-s − 3·11-s + 4·13-s − 6·15-s − 3·17-s − 19-s − 2·21-s + 4·25-s − 4·27-s − 6·29-s − 4·31-s − 6·33-s + 3·35-s − 2·37-s + 8·39-s − 6·41-s + 43-s − 3·45-s − 3·47-s − 6·49-s − 6·51-s − 12·53-s + 9·55-s − 2·57-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.34·5-s − 0.377·7-s + 1/3·9-s − 0.904·11-s + 1.10·13-s − 1.54·15-s − 0.727·17-s − 0.229·19-s − 0.436·21-s + 4/5·25-s − 0.769·27-s − 1.11·29-s − 0.718·31-s − 1.04·33-s + 0.507·35-s − 0.328·37-s + 1.28·39-s − 0.937·41-s + 0.152·43-s − 0.447·45-s − 0.437·47-s − 6/7·49-s − 0.840·51-s − 1.64·53-s + 1.21·55-s − 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−9.035139763634884351001508227661, −8.457332600420497477897703269253, −7.86885586734404145925295723068, −7.15811652768508964807251071197, −6.05813289271022268292942638231, −4.80419327140120764096716079767, −3.67310116806022314895429635564, −3.31935920587958999413340629739, −2.04259190544357015606449170426, 0,
2.04259190544357015606449170426, 3.31935920587958999413340629739, 3.67310116806022314895429635564, 4.80419327140120764096716079767, 6.05813289271022268292942638231, 7.15811652768508964807251071197, 7.86885586734404145925295723068, 8.457332600420497477897703269253, 9.035139763634884351001508227661
