| L(s) = 1 | − 2-s + 4-s + 5-s − 3·7-s − 8-s − 10-s − 13-s + 3·14-s + 16-s − 8·17-s + 5·19-s + 20-s + 2·23-s + 25-s + 26-s − 3·28-s − 4·29-s + 5·31-s − 32-s + 8·34-s − 3·35-s − 5·38-s − 40-s + 2·41-s − 9·43-s − 2·46-s − 47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.13·7-s − 0.353·8-s − 0.316·10-s − 0.277·13-s + 0.801·14-s + 1/4·16-s − 1.94·17-s + 1.14·19-s + 0.223·20-s + 0.417·23-s + 1/5·25-s + 0.196·26-s − 0.566·28-s − 0.742·29-s + 0.898·31-s − 0.176·32-s + 1.37·34-s − 0.507·35-s − 0.811·38-s − 0.158·40-s + 0.312·41-s − 1.37·43-s − 0.294·46-s − 0.145·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 17 | \( 1 + 8 T + p T^{2} \) | 1.17.i |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 9 T + p T^{2} \) | 1.43.j |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + T + p T^{2} \) | 1.59.b |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 - 13 T + p T^{2} \) | 1.89.an |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
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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
−13.60315329549273, −13.14329785592298, −12.74483949769499, −12.20390325844477, −11.56497266178470, −11.24223348212748, −10.68719735047746, −10.12514329000370, −9.729717884734315, −9.325516280048471, −8.919157997365633, −8.480909721158268, −7.651678254727105, −7.351318109560396, −6.662312400192533, −6.336146571637845, −5.997743646756939, −5.052405873581251, −4.766213642704336, −3.928345828190383, −3.136418422034419, −2.934975134138598, −2.099527126484657, −1.612818514944601, −0.6644985801051989, 0,
0.6644985801051989, 1.612818514944601, 2.099527126484657, 2.934975134138598, 3.136418422034419, 3.928345828190383, 4.766213642704336, 5.052405873581251, 5.997743646756939, 6.336146571637845, 6.662312400192533, 7.351318109560396, 7.651678254727105, 8.480909721158268, 8.919157997365633, 9.325516280048471, 9.729717884734315, 10.12514329000370, 10.68719735047746, 11.24223348212748, 11.56497266178470, 12.20390325844477, 12.74483949769499, 13.14329785592298, 13.60315329549273
