| L(s) = 1 | + 2·3-s + 5-s + 4·7-s + 9-s − 11-s + 4·13-s + 2·15-s + 4·17-s + 4·19-s + 8·21-s − 6·23-s + 25-s − 4·27-s − 10·29-s − 4·31-s − 2·33-s + 4·35-s + 2·37-s + 8·39-s − 10·41-s + 8·43-s + 45-s + 6·47-s + 9·49-s + 8·51-s + 2·53-s − 55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s + 0.516·15-s + 0.970·17-s + 0.917·19-s + 1.74·21-s − 1.25·23-s + 1/5·25-s − 0.769·27-s − 1.85·29-s − 0.718·31-s − 0.348·33-s + 0.676·35-s + 0.328·37-s + 1.28·39-s − 1.56·41-s + 1.21·43-s + 0.149·45-s + 0.875·47-s + 9/7·49-s + 1.12·51-s + 0.274·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.366048320\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.366048320\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 12 T + p T^{2} \) | 1.73.am |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 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
−9.171444684967333959644563276068, −8.430504583847850828437166816068, −7.87250663858398564803871887844, −7.32755497663472783830580886838, −5.79585446026787716953342483093, −5.39771920980252886633836422060, −4.09748454363254359064679042672, −3.35779430418487990253666342356, −2.16574614646445950481838552141, −1.43098148196231303000025265209,
1.43098148196231303000025265209, 2.16574614646445950481838552141, 3.35779430418487990253666342356, 4.09748454363254359064679042672, 5.39771920980252886633836422060, 5.79585446026787716953342483093, 7.32755497663472783830580886838, 7.87250663858398564803871887844, 8.430504583847850828437166816068, 9.171444684967333959644563276068
