| L(s) = 1 | − 2·4-s + 11-s + 5·13-s + 4·16-s − 6·17-s − 2·19-s − 3·23-s − 9·29-s + 4·31-s − 2·37-s + 9·41-s − 11·43-s − 2·44-s − 3·47-s − 10·52-s − 3·53-s − 8·61-s − 8·64-s − 8·67-s + 12·68-s + 6·71-s + 14·73-s + 4·76-s − 10·79-s + 12·83-s + 18·89-s + 6·92-s + ⋯ |
| L(s) = 1 | − 4-s + 0.301·11-s + 1.38·13-s + 16-s − 1.45·17-s − 0.458·19-s − 0.625·23-s − 1.67·29-s + 0.718·31-s − 0.328·37-s + 1.40·41-s − 1.67·43-s − 0.301·44-s − 0.437·47-s − 1.38·52-s − 0.412·53-s − 1.02·61-s − 64-s − 0.977·67-s + 1.45·68-s + 0.712·71-s + 1.63·73-s + 0.458·76-s − 1.12·79-s + 1.31·83-s + 1.90·89-s + 0.625·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−13.66102467059513, −13.34761429312491, −13.00061066951716, −12.50606269212545, −11.79781525190163, −11.40547880705358, −10.77999144795198, −10.54252188266615, −9.771955104507336, −9.310570173689319, −8.877352634682839, −8.568027767282288, −7.914989360738793, −7.572443776528111, −6.578351926728436, −6.343214645460221, −5.832749982810460, −5.096260950080874, −4.631890911515905, −4.004257289380808, −3.709271354467166, −3.065576916555283, −2.077870227854911, −1.621078759079481, −0.7304049016532751, 0,
0.7304049016532751, 1.621078759079481, 2.077870227854911, 3.065576916555283, 3.709271354467166, 4.004257289380808, 4.631890911515905, 5.096260950080874, 5.832749982810460, 6.343214645460221, 6.578351926728436, 7.572443776528111, 7.914989360738793, 8.568027767282288, 8.877352634682839, 9.310570173689319, 9.771955104507336, 10.54252188266615, 10.77999144795198, 11.40547880705358, 11.79781525190163, 12.50606269212545, 13.00061066951716, 13.34761429312491, 13.66102467059513
