| L(s) = 1 | − 2·2-s + 2·4-s + 4·7-s + 5·11-s − 6·13-s − 8·14-s − 4·16-s + 5·17-s − 5·19-s − 10·22-s + 23-s + 12·26-s + 8·28-s − 3·29-s + 31-s + 8·32-s − 10·34-s + 2·37-s + 10·38-s − 6·41-s − 6·43-s + 10·44-s − 2·46-s − 2·47-s + 9·49-s − 12·52-s − 3·53-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 1.51·7-s + 1.50·11-s − 1.66·13-s − 2.13·14-s − 16-s + 1.21·17-s − 1.14·19-s − 2.13·22-s + 0.208·23-s + 2.35·26-s + 1.51·28-s − 0.557·29-s + 0.179·31-s + 1.41·32-s − 1.71·34-s + 0.328·37-s + 1.62·38-s − 0.937·41-s − 0.914·43-s + 1.50·44-s − 0.294·46-s − 0.291·47-s + 9/7·49-s − 1.66·52-s − 0.412·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.139996256\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.139996256\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 31 | \( 1 - T \) | |
| good | 2 | \( 1 + p T + p T^{2} \) | 1.2.c |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - T + p T^{2} \) | 1.83.ab |
| 89 | \( 1 - 7 T + p T^{2} \) | 1.89.ah |
| 97 | \( 1 - 3 T + p T^{2} \) | 1.97.ad |
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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
−8.161021127708632613601041998264, −7.39251305224947562546434098391, −6.97142682864840984043928965511, −6.04281997998948684861471474419, −4.90771152390593268237808766963, −4.59967887990570841247796214764, −3.50977181900326279971826472600, −2.13604229208592784510081571551, −1.68474993491626544255421354295, −0.70639465026379569436806460739,
0.70639465026379569436806460739, 1.68474993491626544255421354295, 2.13604229208592784510081571551, 3.50977181900326279971826472600, 4.59967887990570841247796214764, 4.90771152390593268237808766963, 6.04281997998948684861471474419, 6.97142682864840984043928965511, 7.39251305224947562546434098391, 8.161021127708632613601041998264
