| L(s) = 1 | − 2-s + 2·3-s + 4-s + 4·5-s − 2·6-s + 7-s − 8-s + 9-s − 4·10-s − 4·11-s + 2·12-s − 4·13-s − 14-s + 8·15-s + 16-s − 17-s − 18-s − 6·19-s + 4·20-s + 2·21-s + 4·22-s − 2·24-s + 11·25-s + 4·26-s − 4·27-s + 28-s + 6·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s + 1.78·5-s − 0.816·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 1.26·10-s − 1.20·11-s + 0.577·12-s − 1.10·13-s − 0.267·14-s + 2.06·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 1.37·19-s + 0.894·20-s + 0.436·21-s + 0.852·22-s − 0.408·24-s + 11/5·25-s + 0.784·26-s − 0.769·27-s + 0.188·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 238 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 238 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.521941528\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.521941528\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 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
−12.30264776476448446316558003589, −10.56616726919396924043112650950, −10.17628989538336803184934071125, −9.124388696960328040722462575539, −8.504258525876890475087699972026, −7.42012488546349064955813040971, −6.15855370813537923128983581072, −4.95404247889329802285418964237, −2.66188663960327860854980555616, −2.11300277285141814964879652826,
2.11300277285141814964879652826, 2.66188663960327860854980555616, 4.95404247889329802285418964237, 6.15855370813537923128983581072, 7.42012488546349064955813040971, 8.504258525876890475087699972026, 9.124388696960328040722462575539, 10.17628989538336803184934071125, 10.56616726919396924043112650950, 12.30264776476448446316558003589
