| L(s) = 1 | + 2·5-s + 2·7-s − 17-s − 2·19-s − 2·23-s − 25-s + 10·29-s + 8·31-s + 4·35-s − 6·37-s + 2·41-s − 2·43-s − 3·49-s + 12·53-s − 10·59-s − 10·61-s − 4·67-s − 6·71-s + 10·79-s − 4·83-s − 2·85-s + 10·89-s − 4·95-s − 14·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.755·7-s − 0.242·17-s − 0.458·19-s − 0.417·23-s − 1/5·25-s + 1.85·29-s + 1.43·31-s + 0.676·35-s − 0.986·37-s + 0.312·41-s − 0.304·43-s − 3/7·49-s + 1.64·53-s − 1.30·59-s − 1.28·61-s − 0.488·67-s − 0.712·71-s + 1.12·79-s − 0.439·83-s − 0.216·85-s + 1.05·89-s − 0.410·95-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 148104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 148104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 11 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−13.60846208374603, −13.38270952712041, −12.49131116037208, −12.20120325842017, −11.75617710216884, −11.19462324613105, −10.60994973762156, −10.17857295391047, −9.964849387023205, −9.190556438919811, −8.736950241689572, −8.331273885950734, −7.821200773994078, −7.252267029079879, −6.559801680403236, −6.191390994450575, −5.795154091885851, −4.895961813175433, −4.797546801334067, −4.134993214322514, −3.381172179092251, −2.658482097162713, −2.235016960078651, −1.545258277800815, −1.032885967990084, 0,
1.032885967990084, 1.545258277800815, 2.235016960078651, 2.658482097162713, 3.381172179092251, 4.134993214322514, 4.797546801334067, 4.895961813175433, 5.795154091885851, 6.191390994450575, 6.559801680403236, 7.252267029079879, 7.821200773994078, 8.331273885950734, 8.736950241689572, 9.190556438919811, 9.964849387023205, 10.17857295391047, 10.60994973762156, 11.19462324613105, 11.75617710216884, 12.20120325842017, 12.49131116037208, 13.38270952712041, 13.60846208374603
