L(s) = 1 | − 5-s + 2·11-s − 13-s + 2·17-s + 7·19-s − 6·23-s + 25-s − 4·29-s − 3·31-s − 2·37-s − 8·41-s − 11·43-s + 4·47-s − 7·49-s − 2·53-s − 2·55-s − 10·59-s + 5·61-s + 65-s − 15·67-s − 10·71-s + 73-s − 15·79-s + 6·83-s − 2·85-s + 12·89-s − 7·95-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.603·11-s − 0.277·13-s + 0.485·17-s + 1.60·19-s − 1.25·23-s + 1/5·25-s − 0.742·29-s − 0.538·31-s − 0.328·37-s − 1.24·41-s − 1.67·43-s + 0.583·47-s − 49-s − 0.274·53-s − 0.269·55-s − 1.30·59-s + 0.640·61-s + 0.124·65-s − 1.83·67-s − 1.18·71-s + 0.117·73-s − 1.68·79-s + 0.658·83-s − 0.216·85-s + 1.27·89-s − 0.718·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.271071390\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.271071390\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 15 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−14.06357257906755, −13.59370487418863, −13.05671554116245, −12.41702613681176, −11.89144346141905, −11.68185675018364, −11.23216866758024, −10.35400712492424, −10.04510376618307, −9.556426341673288, −8.930560366961155, −8.497907272876898, −7.742718674872198, −7.463193186298261, −6.970089039223084, −6.155382225533408, −5.819540701747671, −4.995587534283407, −4.699742848154650, −3.743816947005628, −3.461667193545963, −2.877324288333733, −1.788050265451508, −1.458381970854047, −0.3664972267784199,
0.3664972267784199, 1.458381970854047, 1.788050265451508, 2.877324288333733, 3.461667193545963, 3.743816947005628, 4.699742848154650, 4.995587534283407, 5.819540701747671, 6.155382225533408, 6.970089039223084, 7.463193186298261, 7.742718674872198, 8.497907272876898, 8.930560366961155, 9.556426341673288, 10.04510376618307, 10.35400712492424, 11.23216866758024, 11.68185675018364, 11.89144346141905, 12.41702613681176, 13.05671554116245, 13.59370487418863, 14.06357257906755