| L(s) = 1 | + 2-s + 3-s − 4-s + 6-s + 3·7-s − 3·8-s + 9-s − 12-s + 3·14-s − 16-s − 6·17-s + 18-s − 19-s + 3·21-s − 3·23-s − 3·24-s + 27-s − 3·28-s + 6·29-s − 10·31-s + 5·32-s − 6·34-s − 36-s + 9·37-s − 38-s + 10·41-s + 3·42-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s + 1.13·7-s − 1.06·8-s + 1/3·9-s − 0.288·12-s + 0.801·14-s − 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.229·19-s + 0.654·21-s − 0.625·23-s − 0.612·24-s + 0.192·27-s − 0.566·28-s + 1.11·29-s − 1.79·31-s + 0.883·32-s − 1.02·34-s − 1/6·36-s + 1.47·37-s − 0.162·38-s + 1.56·41-s + 0.462·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.144738224\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.144738224\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 9 T + p T^{2} \) | 1.37.aj |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 9 T + p T^{2} \) | 1.61.aj |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−12.92282535491213, −12.61814438075025, −12.11506890570188, −11.37896479432832, −11.25781563187964, −10.62597032737995, −10.14393186173431, −9.432784233427887, −8.983633376384012, −8.859328450335171, −8.164076970651085, −7.782000095399233, −7.308950525234171, −6.638468494526013, −6.024966542585949, −5.683538142277986, −5.000894469813766, −4.503565628672915, −4.186475387716444, −3.835367463873555, −3.003813260656709, −2.348041521600632, −2.131752581622367, −1.164328103076719, −0.5045323667971182,
0.5045323667971182, 1.164328103076719, 2.131752581622367, 2.348041521600632, 3.003813260656709, 3.835367463873555, 4.186475387716444, 4.503565628672915, 5.000894469813766, 5.683538142277986, 6.024966542585949, 6.638468494526013, 7.308950525234171, 7.782000095399233, 8.164076970651085, 8.859328450335171, 8.983633376384012, 9.432784233427887, 10.14393186173431, 10.62597032737995, 11.25781563187964, 11.37896479432832, 12.11506890570188, 12.61814438075025, 12.92282535491213
