| L(s) = 1 | − 2·3-s + 9-s − 6·11-s + 2·19-s − 5·25-s + 4·27-s + 12·33-s − 6·41-s − 10·43-s − 7·49-s − 4·57-s + 6·59-s − 14·67-s + 2·73-s + 10·75-s − 11·81-s + 18·83-s − 18·89-s − 10·97-s − 6·99-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/3·9-s − 1.80·11-s + 0.458·19-s − 25-s + 0.769·27-s + 2.08·33-s − 0.937·41-s − 1.52·43-s − 49-s − 0.529·57-s + 0.781·59-s − 1.71·67-s + 0.234·73-s + 1.15·75-s − 1.22·81-s + 1.97·83-s − 1.90·89-s − 1.01·97-s − 0.603·99-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 18 T + p T^{2} \) | 1.83.as |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 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
−14.73429153885774, −13.84520359535265, −13.52422733402692, −13.09086806301249, −12.48641679259496, −12.05391976594045, −11.48494403939372, −11.21354326658703, −10.49210466321803, −10.23951090410705, −9.768728402012127, −9.041196435197200, −8.278811896301974, −7.979604561043147, −7.394476057733181, −6.717066722035525, −6.276568114978124, −5.540189063927464, −5.275429788536788, −4.864019878134659, −4.106975683867754, −3.272153532839683, −2.762029878943754, −1.979327451415643, −1.182099822668794, 0, 0,
1.182099822668794, 1.979327451415643, 2.762029878943754, 3.272153532839683, 4.106975683867754, 4.864019878134659, 5.275429788536788, 5.540189063927464, 6.276568114978124, 6.717066722035525, 7.394476057733181, 7.979604561043147, 8.278811896301974, 9.041196435197200, 9.768728402012127, 10.23951090410705, 10.49210466321803, 11.21354326658703, 11.48494403939372, 12.05391976594045, 12.48641679259496, 13.09086806301249, 13.52422733402692, 13.84520359535265, 14.73429153885774
