| L(s) = 1 | + 3-s − 2·5-s + 9-s − 13-s − 2·15-s + 6·17-s − 8·19-s − 8·23-s − 25-s + 27-s − 2·29-s − 8·31-s − 10·37-s − 39-s − 6·41-s − 4·43-s − 2·45-s − 7·49-s + 6·51-s − 14·53-s − 8·57-s + 12·59-s + 10·61-s + 2·65-s + 8·67-s − 8·69-s + 14·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s − 0.277·13-s − 0.516·15-s + 1.45·17-s − 1.83·19-s − 1.66·23-s − 1/5·25-s + 0.192·27-s − 0.371·29-s − 1.43·31-s − 1.64·37-s − 0.160·39-s − 0.937·41-s − 0.609·43-s − 0.298·45-s − 49-s + 0.840·51-s − 1.92·53-s − 1.05·57-s + 1.56·59-s + 1.28·61-s + 0.248·65-s + 0.977·67-s − 0.963·69-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−13.96204893304236, −13.26113635440503, −12.77218518454788, −12.38329148466785, −12.05828478006805, −11.40719559407462, −11.03291264755771, −10.36656185150420, −9.922337548490134, −9.641429755488211, −8.827679994127359, −8.318842658795199, −8.119254887284217, −7.622891485732052, −7.047183077412103, −6.552952360220034, −5.972954171787956, −5.241068882272588, −4.886577842329342, −3.983949344335300, −3.654788451941504, −3.475318896340354, −2.395928491241540, −1.972777445432613, −1.352095544682272, 0, 0,
1.352095544682272, 1.972777445432613, 2.395928491241540, 3.475318896340354, 3.654788451941504, 3.983949344335300, 4.886577842329342, 5.241068882272588, 5.972954171787956, 6.552952360220034, 7.047183077412103, 7.622891485732052, 8.119254887284217, 8.318842658795199, 8.827679994127359, 9.641429755488211, 9.922337548490134, 10.36656185150420, 11.03291264755771, 11.40719559407462, 12.05828478006805, 12.38329148466785, 12.77218518454788, 13.26113635440503, 13.96204893304236
