| L(s) = 1 | + 4·5-s + 11-s − 4·13-s − 6·17-s − 2·19-s + 11·25-s − 10·29-s + 8·31-s − 6·37-s + 2·41-s + 12·43-s − 8·47-s − 6·53-s + 4·55-s + 14·59-s − 16·65-s − 8·67-s + 8·71-s + 10·73-s − 8·79-s + 10·83-s − 24·85-s + 2·89-s − 8·95-s − 14·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 0.301·11-s − 1.10·13-s − 1.45·17-s − 0.458·19-s + 11/5·25-s − 1.85·29-s + 1.43·31-s − 0.986·37-s + 0.312·41-s + 1.82·43-s − 1.16·47-s − 0.824·53-s + 0.539·55-s + 1.82·59-s − 1.98·65-s − 0.977·67-s + 0.949·71-s + 1.17·73-s − 0.900·79-s + 1.09·83-s − 2.60·85-s + 0.211·89-s − 0.820·95-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 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 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 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
−13.51278857644239, −13.12196437891840, −12.68991855973003, −12.32630043140757, −11.53129265028854, −11.11092358988168, −10.66211268111879, −10.00964948452955, −9.806639372912730, −9.203338145328033, −8.962552043250537, −8.374338047567314, −7.640351596564464, −7.055650058385636, −6.632442827431391, −6.167022877667642, −5.720197595227592, −5.082353193132937, −4.722925980872749, −4.097724978181680, −3.327227022477138, −2.497519910974146, −2.220263596282261, −1.783790495965460, −0.9232420084419944, 0,
0.9232420084419944, 1.783790495965460, 2.220263596282261, 2.497519910974146, 3.327227022477138, 4.097724978181680, 4.722925980872749, 5.082353193132937, 5.720197595227592, 6.167022877667642, 6.632442827431391, 7.055650058385636, 7.640351596564464, 8.374338047567314, 8.962552043250537, 9.203338145328033, 9.806639372912730, 10.00964948452955, 10.66211268111879, 11.11092358988168, 11.53129265028854, 12.32630043140757, 12.68991855973003, 13.12196437891840, 13.51278857644239
