| L(s) = 1 | − 7-s + 2·11-s − 13-s − 4·17-s + 4·19-s − 2·23-s − 5·25-s − 6·29-s + 10·37-s − 4·41-s − 4·43-s + 49-s − 10·53-s − 4·59-s − 2·61-s + 6·71-s + 14·73-s − 2·77-s − 4·79-s − 16·83-s − 12·89-s + 91-s + 6·97-s + 8·103-s − 18·107-s − 10·109-s + 14·113-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 0.603·11-s − 0.277·13-s − 0.970·17-s + 0.917·19-s − 0.417·23-s − 25-s − 1.11·29-s + 1.64·37-s − 0.624·41-s − 0.609·43-s + 1/7·49-s − 1.37·53-s − 0.520·59-s − 0.256·61-s + 0.712·71-s + 1.63·73-s − 0.227·77-s − 0.450·79-s − 1.75·83-s − 1.27·89-s + 0.104·91-s + 0.609·97-s + 0.788·103-s − 1.74·107-s − 0.957·109-s + 1.31·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 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
−8.187779556583543071802035119636, −7.56386095768078836949457966906, −6.71701075830552720669434017289, −6.09510035748296873349035012747, −5.24551383519126216163915722891, −4.29345851534004351357225561275, −3.58004430162338775504314652387, −2.55231147487554452797225259181, −1.50974998810809266451094332019, 0,
1.50974998810809266451094332019, 2.55231147487554452797225259181, 3.58004430162338775504314652387, 4.29345851534004351357225561275, 5.24551383519126216163915722891, 6.09510035748296873349035012747, 6.71701075830552720669434017289, 7.56386095768078836949457966906, 8.187779556583543071802035119636
