| L(s) = 1 | − 3-s − 4·5-s + 9-s + 4·15-s − 2·17-s − 4·23-s + 11·25-s − 27-s + 2·29-s − 8·31-s + 8·37-s + 4·41-s + 8·43-s − 4·45-s + 2·51-s + 6·53-s + 8·59-s + 10·61-s + 8·67-s + 4·69-s + 4·73-s − 11·75-s + 12·79-s + 81-s + 8·83-s + 8·85-s − 2·87-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.78·5-s + 1/3·9-s + 1.03·15-s − 0.485·17-s − 0.834·23-s + 11/5·25-s − 0.192·27-s + 0.371·29-s − 1.43·31-s + 1.31·37-s + 0.624·41-s + 1.21·43-s − 0.596·45-s + 0.280·51-s + 0.824·53-s + 1.04·59-s + 1.28·61-s + 0.977·67-s + 0.481·69-s + 0.468·73-s − 1.27·75-s + 1.35·79-s + 1/9·81-s + 0.878·83-s + 0.867·85-s − 0.214·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99372 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99372 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.296734651\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.296734651\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 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
−13.78057808411161, −12.93870266582614, −12.79091468259887, −12.22542660834407, −11.69102147922974, −11.42194621565772, −10.91924975347544, −10.56884805064973, −9.846285494940056, −9.267383956756431, −8.690683294365890, −8.203291362694888, −7.604697338665040, −7.390858138768003, −6.727069631245778, −6.204403230030282, −5.550592080279650, −4.917297168271017, −4.405273751156251, −3.774550357301377, −3.658934795960179, −2.604860584887498, −2.041839915258147, −0.8152837555480384, −0.5320334257880803,
0.5320334257880803, 0.8152837555480384, 2.041839915258147, 2.604860584887498, 3.658934795960179, 3.774550357301377, 4.405273751156251, 4.917297168271017, 5.550592080279650, 6.204403230030282, 6.727069631245778, 7.390858138768003, 7.604697338665040, 8.203291362694888, 8.690683294365890, 9.267383956756431, 9.846285494940056, 10.56884805064973, 10.91924975347544, 11.42194621565772, 11.69102147922974, 12.22542660834407, 12.79091468259887, 12.93870266582614, 13.78057808411161
