| L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s − 3·11-s − 13-s − 14-s + 16-s − 3·17-s + 2·19-s − 20-s − 3·22-s − 3·23-s + 25-s − 26-s − 28-s + 3·29-s − 4·31-s + 32-s − 3·34-s + 35-s + 8·37-s + 2·38-s − 40-s + 6·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 0.316·10-s − 0.904·11-s − 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.727·17-s + 0.458·19-s − 0.223·20-s − 0.639·22-s − 0.625·23-s + 1/5·25-s − 0.196·26-s − 0.188·28-s + 0.557·29-s − 0.718·31-s + 0.176·32-s − 0.514·34-s + 0.169·35-s + 1.31·37-s + 0.324·38-s − 0.158·40-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.239285214\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.239285214\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 79 | \( 1 - T \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 13 T + p T^{2} \) | 1.97.n |
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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
−7.902171118955201340644018299202, −7.14968516223648091601454857204, −6.51762281483711566152924071010, −5.74574047823770243817881281733, −5.06889689922587362060341587115, −4.34192381547350858819925483891, −3.64600695081610433319420464262, −2.77050340123471431503908003294, −2.13645972607153276692200913063, −0.65388908611375751714224477477,
0.65388908611375751714224477477, 2.13645972607153276692200913063, 2.77050340123471431503908003294, 3.64600695081610433319420464262, 4.34192381547350858819925483891, 5.06889689922587362060341587115, 5.74574047823770243817881281733, 6.51762281483711566152924071010, 7.14968516223648091601454857204, 7.902171118955201340644018299202
