| L(s) = 1 | + 5-s + 4·7-s − 4·11-s − 5·13-s + 17-s + 3·19-s + 8·23-s − 4·25-s − 7·31-s + 4·35-s − 4·37-s − 41-s + 6·43-s + 8·47-s + 9·49-s + 14·53-s − 4·55-s + 7·59-s − 8·61-s − 5·65-s + 5·67-s + 7·71-s + 73-s − 16·77-s + 14·79-s + 15·83-s + 85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s − 1.20·11-s − 1.38·13-s + 0.242·17-s + 0.688·19-s + 1.66·23-s − 4/5·25-s − 1.25·31-s + 0.676·35-s − 0.657·37-s − 0.156·41-s + 0.914·43-s + 1.16·47-s + 9/7·49-s + 1.92·53-s − 0.539·55-s + 0.911·59-s − 1.02·61-s − 0.620·65-s + 0.610·67-s + 0.830·71-s + 0.117·73-s − 1.82·77-s + 1.57·79-s + 1.64·83-s + 0.108·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.346231852\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.346231852\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 41 | \( 1 + T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 - 7 T + p T^{2} \) | 1.59.ah |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 - 15 T + p T^{2} \) | 1.83.ap |
| 89 | \( 1 - 13 T + p T^{2} \) | 1.89.an |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−7.75479609638146414636337103803, −7.64963861441685054579950570469, −6.89640970510413981108740082338, −5.55845618691973608426218003834, −5.25436541274672898817674287592, −4.76691307230969803304851278592, −3.65181400119607815214866639941, −2.51169692252237383369464094315, −2.03482712180999266985598123995, −0.809760234295985811376338829405,
0.809760234295985811376338829405, 2.03482712180999266985598123995, 2.51169692252237383369464094315, 3.65181400119607815214866639941, 4.76691307230969803304851278592, 5.25436541274672898817674287592, 5.55845618691973608426218003834, 6.89640970510413981108740082338, 7.64963861441685054579950570469, 7.75479609638146414636337103803
