| L(s) = 1 | − 4·5-s + 4·7-s + 2·11-s + 13-s + 6·17-s + 4·19-s + 4·23-s + 11·25-s − 6·29-s − 8·31-s − 16·35-s + 10·37-s + 4·41-s − 4·43-s − 6·47-s + 9·49-s + 6·53-s − 8·55-s + 6·59-s + 6·61-s − 4·65-s + 10·71-s − 2·73-s + 8·77-s + 10·83-s − 24·85-s − 8·89-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 1.51·7-s + 0.603·11-s + 0.277·13-s + 1.45·17-s + 0.917·19-s + 0.834·23-s + 11/5·25-s − 1.11·29-s − 1.43·31-s − 2.70·35-s + 1.64·37-s + 0.624·41-s − 0.609·43-s − 0.875·47-s + 9/7·49-s + 0.824·53-s − 1.07·55-s + 0.781·59-s + 0.768·61-s − 0.496·65-s + 1.18·71-s − 0.234·73-s + 0.911·77-s + 1.09·83-s − 2.60·85-s − 0.847·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.045885049\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.045885049\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−7.84299131779587178569895671297, −7.46342453313584230019118360575, −6.78089096086834908852686039810, −5.46319754519636655265339925087, −5.14662523509473106904036588761, −4.09435176299410547015277416426, −3.79256094813629132082947144890, −2.89877338445118556705012658271, −1.51493457273732862172450204099, −0.798927434561717906550639137243,
0.798927434561717906550639137243, 1.51493457273732862172450204099, 2.89877338445118556705012658271, 3.79256094813629132082947144890, 4.09435176299410547015277416426, 5.14662523509473106904036588761, 5.46319754519636655265339925087, 6.78089096086834908852686039810, 7.46342453313584230019118360575, 7.84299131779587178569895671297
