| L(s) = 1 | + 5-s − 3·9-s + 3·11-s − 5·17-s + 5·19-s + 25-s − 8·29-s + 5·31-s + 10·37-s − 4·41-s − 43-s − 3·45-s + 7·47-s − 3·53-s + 3·55-s + 12·59-s − 6·61-s + 2·67-s − 6·71-s − 8·73-s − 79-s + 9·81-s − 5·85-s + 15·89-s + 5·95-s − 10·97-s − 9·99-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 9-s + 0.904·11-s − 1.21·17-s + 1.14·19-s + 1/5·25-s − 1.48·29-s + 0.898·31-s + 1.64·37-s − 0.624·41-s − 0.152·43-s − 0.447·45-s + 1.02·47-s − 0.412·53-s + 0.404·55-s + 1.56·59-s − 0.768·61-s + 0.244·67-s − 0.712·71-s − 0.936·73-s − 0.112·79-s + 81-s − 0.542·85-s + 1.58·89-s + 0.512·95-s − 1.01·97-s − 0.904·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 79 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−13.08138497921478, −12.27549228716116, −11.92766158392279, −11.51096217184413, −11.09507569135165, −10.78850824451120, −10.02591096617257, −9.554621695557123, −9.300754744269774, −8.765055135662667, −8.417709389480123, −7.774575862419463, −7.301662400629148, −6.750122270407048, −6.294525347647261, −5.846249851856920, −5.424240544270212, −4.863684042049094, −4.226960903278738, −3.825141650985068, −3.103330847765235, −2.657404711987519, −2.110067821537947, −1.428000057582949, −0.8028680512668185, 0,
0.8028680512668185, 1.428000057582949, 2.110067821537947, 2.657404711987519, 3.103330847765235, 3.825141650985068, 4.226960903278738, 4.863684042049094, 5.424240544270212, 5.846249851856920, 6.294525347647261, 6.750122270407048, 7.301662400629148, 7.774575862419463, 8.417709389480123, 8.765055135662667, 9.300754744269774, 9.554621695557123, 10.02591096617257, 10.78850824451120, 11.09507569135165, 11.51096217184413, 11.92766158392279, 12.27549228716116, 13.08138497921478
