| L(s) = 1 | − 4·7-s − 4·11-s + 13-s − 6·17-s + 8·23-s + 6·29-s + 4·31-s − 2·37-s + 10·41-s + 4·43-s + 8·47-s + 9·49-s + 2·53-s − 12·59-s + 2·61-s + 16·67-s − 8·71-s + 6·73-s + 16·77-s + 16·79-s + 4·83-s + 2·89-s − 4·91-s − 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 1.20·11-s + 0.277·13-s − 1.45·17-s + 1.66·23-s + 1.11·29-s + 0.718·31-s − 0.328·37-s + 1.56·41-s + 0.609·43-s + 1.16·47-s + 9/7·49-s + 0.274·53-s − 1.56·59-s + 0.256·61-s + 1.95·67-s − 0.949·71-s + 0.702·73-s + 1.82·77-s + 1.80·79-s + 0.439·83-s + 0.211·89-s − 0.419·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.186684000\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.186684000\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 16 T + p T^{2} \) | 1.67.aq |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−12.92609491993775, −12.86751361197352, −12.37340001667976, −11.76343137925251, −11.01898680765039, −10.76849903379840, −10.43192825680109, −9.707653910279331, −9.409811863225260, −8.807106416716624, −8.551868543138245, −7.784662758067021, −7.247923011493089, −6.876343526055751, −6.232805321382046, −6.047894743618513, −5.266260839044361, −4.689411379864849, −4.327317577193479, −3.428691505078544, −3.109327857709511, −2.473044476237563, −2.142668493168199, −0.7632767976873249, −0.6058238944390383,
0.6058238944390383, 0.7632767976873249, 2.142668493168199, 2.473044476237563, 3.109327857709511, 3.428691505078544, 4.327317577193479, 4.689411379864849, 5.266260839044361, 6.047894743618513, 6.232805321382046, 6.876343526055751, 7.247923011493089, 7.784662758067021, 8.551868543138245, 8.807106416716624, 9.409811863225260, 9.707653910279331, 10.43192825680109, 10.76849903379840, 11.01898680765039, 11.76343137925251, 12.37340001667976, 12.86751361197352, 12.92609491993775
