| L(s) = 1 | − 3-s + 2·5-s + 9-s + 4·11-s − 2·13-s − 2·15-s + 6·17-s + 19-s − 25-s − 27-s − 10·29-s + 8·31-s − 4·33-s − 2·37-s + 2·39-s − 6·41-s − 4·43-s + 2·45-s − 8·47-s − 6·51-s + 6·53-s + 8·55-s − 57-s − 4·59-s + 10·61-s − 4·65-s − 8·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.516·15-s + 1.45·17-s + 0.229·19-s − 1/5·25-s − 0.192·27-s − 1.85·29-s + 1.43·31-s − 0.696·33-s − 0.328·37-s + 0.320·39-s − 0.937·41-s − 0.609·43-s + 0.298·45-s − 1.16·47-s − 0.840·51-s + 0.824·53-s + 1.07·55-s − 0.132·57-s − 0.520·59-s + 1.28·61-s − 0.496·65-s − 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44688 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 19 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−14.88200467330003, −14.36675793973701, −13.90958649170243, −13.34576461425909, −12.91474954550532, −12.08301223187036, −11.92248945009487, −11.44080483370212, −10.68840103448215, −10.07688391778665, −9.688355327293514, −9.439124078587726, −8.607101624401666, −8.004159166560160, −7.321298842379672, −6.806279733588987, −6.246566277134628, −5.679538425723471, −5.281956984737790, −4.622313188641969, −3.803966824451326, −3.324325471108684, −2.386637622302613, −1.593784312272460, −1.162084855780186, 0,
1.162084855780186, 1.593784312272460, 2.386637622302613, 3.324325471108684, 3.803966824451326, 4.622313188641969, 5.281956984737790, 5.679538425723471, 6.246566277134628, 6.806279733588987, 7.321298842379672, 8.004159166560160, 8.607101624401666, 9.439124078587726, 9.688355327293514, 10.07688391778665, 10.68840103448215, 11.44080483370212, 11.92248945009487, 12.08301223187036, 12.91474954550532, 13.34576461425909, 13.90958649170243, 14.36675793973701, 14.88200467330003
