| L(s) = 1 | − 2·2-s + 3-s + 2·4-s − 2·6-s − 2·7-s − 2·9-s − 3·11-s + 2·12-s − 2·13-s + 4·14-s − 4·16-s + 4·17-s + 4·18-s − 4·19-s − 2·21-s + 6·22-s + 4·23-s − 5·25-s + 4·26-s − 5·27-s − 4·28-s − 9·29-s + 8·31-s + 8·32-s − 3·33-s − 8·34-s − 4·36-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s − 0.816·6-s − 0.755·7-s − 2/3·9-s − 0.904·11-s + 0.577·12-s − 0.554·13-s + 1.06·14-s − 16-s + 0.970·17-s + 0.942·18-s − 0.917·19-s − 0.436·21-s + 1.27·22-s + 0.834·23-s − 25-s + 0.784·26-s − 0.962·27-s − 0.755·28-s − 1.67·29-s + 1.43·31-s + 1.41·32-s − 0.522·33-s − 1.37·34-s − 2/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 347 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 347 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 347 | \( 1 + T \) | |
| good | 2 | \( 1 + p T + p T^{2} \) | 1.2.c |
| 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 12 T + p T^{2} \) | 1.37.m |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 11 T + p T^{2} \) | 1.67.l |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 - T + p T^{2} \) | 1.89.ab |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−10.67286158233625109231137184338, −9.863695884755277760497173433101, −9.248978287515319430740698615916, −8.243339481383888265280422341752, −7.70693812613064419776717484552, −6.55737249907312816467319435845, −5.19897604598426877220830062859, −3.39055671180264983859343570689, −2.16919913228947071615788149785, 0,
2.16919913228947071615788149785, 3.39055671180264983859343570689, 5.19897604598426877220830062859, 6.55737249907312816467319435845, 7.70693812613064419776717484552, 8.243339481383888265280422341752, 9.248978287515319430740698615916, 9.863695884755277760497173433101, 10.67286158233625109231137184338
