| L(s) = 1 | − 3-s + 5-s + 9-s + 2·11-s − 4·13-s − 15-s + 2·17-s + 2·23-s + 25-s − 27-s + 6·29-s + 4·31-s − 2·33-s − 10·37-s + 4·39-s − 2·41-s + 4·43-s + 45-s − 8·47-s − 2·51-s + 10·53-s + 2·55-s + 4·59-s + 8·61-s − 4·65-s + 8·67-s − 2·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.603·11-s − 1.10·13-s − 0.258·15-s + 0.485·17-s + 0.417·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.718·31-s − 0.348·33-s − 1.64·37-s + 0.640·39-s − 0.312·41-s + 0.609·43-s + 0.149·45-s − 1.16·47-s − 0.280·51-s + 1.37·53-s + 0.269·55-s + 0.520·59-s + 1.02·61-s − 0.496·65-s + 0.977·67-s − 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.614546281\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.614546281\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 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 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−8.766478863443302363408410975591, −8.021252076005015374509882583726, −6.90842479379130569242180062632, −6.69405569392998382511054784833, −5.54755934934665382318998592213, −5.05253561513726568937694635743, −4.14381992805715318785313309926, −3.05452872527634701154164130670, −2.00146175035894616251096190421, −0.814581355796936259209215958798,
0.814581355796936259209215958798, 2.00146175035894616251096190421, 3.05452872527634701154164130670, 4.14381992805715318785313309926, 5.05253561513726568937694635743, 5.54755934934665382318998592213, 6.69405569392998382511054784833, 6.90842479379130569242180062632, 8.021252076005015374509882583726, 8.766478863443302363408410975591
