| L(s) = 1 | − 3-s − 4·7-s + 9-s + 2·13-s − 2·17-s + 4·19-s + 4·21-s − 27-s − 2·29-s − 8·31-s + 2·37-s − 2·39-s + 10·41-s − 4·43-s + 9·49-s + 2·51-s + 6·53-s − 4·57-s + 12·59-s − 2·61-s − 4·63-s − 12·67-s − 16·71-s − 10·73-s + 4·79-s + 81-s + 2·87-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s + 0.554·13-s − 0.485·17-s + 0.917·19-s + 0.872·21-s − 0.192·27-s − 0.371·29-s − 1.43·31-s + 0.328·37-s − 0.320·39-s + 1.56·41-s − 0.609·43-s + 9/7·49-s + 0.280·51-s + 0.824·53-s − 0.529·57-s + 1.56·59-s − 0.256·61-s − 0.503·63-s − 1.46·67-s − 1.89·71-s − 1.17·73-s + 0.450·79-s + 1/9·81-s + 0.214·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 317400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 317400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4834971059\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4834971059\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.72673812433141, −12.16154904870512, −11.73418912297915, −11.31170838587878, −10.76425830505191, −10.42031833771245, −9.864756357650455, −9.483792180387833, −9.028329536772274, −8.711754114196924, −7.891182244236306, −7.371237656094209, −7.035071998947284, −6.546629941607329, −6.004808701571733, −5.663643265080427, −5.282231724224303, −4.427337183930577, −3.990847262975057, −3.550404191625297, −2.916417280710592, −2.500402423227632, −1.599324776453867, −1.046989767060511, −0.2113620830817361,
0.2113620830817361, 1.046989767060511, 1.599324776453867, 2.500402423227632, 2.916417280710592, 3.550404191625297, 3.990847262975057, 4.427337183930577, 5.282231724224303, 5.663643265080427, 6.004808701571733, 6.546629941607329, 7.035071998947284, 7.371237656094209, 7.891182244236306, 8.711754114196924, 9.028329536772274, 9.483792180387833, 9.864756357650455, 10.42031833771245, 10.76425830505191, 11.31170838587878, 11.73418912297915, 12.16154904870512, 12.72673812433141
