| L(s) = 1 | + 3-s + 9-s + 2·11-s + 5·13-s − 2·17-s + 3·19-s + 2·23-s − 5·25-s + 27-s + 8·29-s − 31-s + 2·33-s − 5·37-s + 5·39-s + 2·41-s + 7·43-s − 8·47-s − 2·51-s − 2·53-s + 3·57-s + 10·59-s − 2·61-s − 11·67-s + 2·69-s + 12·71-s − 3·73-s − 5·75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 0.603·11-s + 1.38·13-s − 0.485·17-s + 0.688·19-s + 0.417·23-s − 25-s + 0.192·27-s + 1.48·29-s − 0.179·31-s + 0.348·33-s − 0.821·37-s + 0.800·39-s + 0.312·41-s + 1.06·43-s − 1.16·47-s − 0.280·51-s − 0.274·53-s + 0.397·57-s + 1.30·59-s − 0.256·61-s − 1.34·67-s + 0.240·69-s + 1.42·71-s − 0.351·73-s − 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.831483071\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.831483071\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 11 T + p T^{2} \) | 1.67.l |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 3 T + p T^{2} \) | 1.73.d |
| 79 | \( 1 - 17 T + p T^{2} \) | 1.79.ar |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 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
−8.367254419830641377161703660648, −7.69740722412821590639966490907, −6.79583497273290358368092666109, −6.26878210072653759587916403007, −5.38308727445969015201591310970, −4.40613304231564838164668855941, −3.71111468413452700110539104489, −2.99707726507203693173786359038, −1.89060552166772460253051663105, −0.962959608137828990973445692205,
0.962959608137828990973445692205, 1.89060552166772460253051663105, 2.99707726507203693173786359038, 3.71111468413452700110539104489, 4.40613304231564838164668855941, 5.38308727445969015201591310970, 6.26878210072653759587916403007, 6.79583497273290358368092666109, 7.69740722412821590639966490907, 8.367254419830641377161703660648
