L(s) = 1 | + 1.56·3-s + 2.56·5-s + 3·7-s − 0.561·9-s − 0.561·11-s + 1.56·13-s + 4·15-s + 0.123·17-s − 19-s + 4.68·21-s + 5.56·23-s + 1.56·25-s − 5.56·27-s − 4.68·29-s + 9.12·31-s − 0.876·33-s + 7.68·35-s − 7.12·37-s + 2.43·39-s + 4·41-s − 5.43·43-s − 1.43·45-s − 3.68·47-s + 2·49-s + 0.192·51-s + 4.43·53-s − 1.43·55-s + ⋯ |
L(s) = 1 | + 0.901·3-s + 1.14·5-s + 1.13·7-s − 0.187·9-s − 0.169·11-s + 0.433·13-s + 1.03·15-s + 0.0298·17-s − 0.229·19-s + 1.02·21-s + 1.15·23-s + 0.312·25-s − 1.07·27-s − 0.869·29-s + 1.63·31-s − 0.152·33-s + 1.29·35-s − 1.17·37-s + 0.390·39-s + 0.624·41-s − 0.829·43-s − 0.214·45-s − 0.537·47-s + 0.285·49-s + 0.0269·51-s + 0.609·53-s − 0.193·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.969683653\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.969683653\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 - 1.56T + 3T^{2} \) |
| 5 | \( 1 - 2.56T + 5T^{2} \) |
| 7 | \( 1 - 3T + 7T^{2} \) |
| 11 | \( 1 + 0.561T + 11T^{2} \) |
| 13 | \( 1 - 1.56T + 13T^{2} \) |
| 17 | \( 1 - 0.123T + 17T^{2} \) |
| 23 | \( 1 - 5.56T + 23T^{2} \) |
| 29 | \( 1 + 4.68T + 29T^{2} \) |
| 31 | \( 1 - 9.12T + 31T^{2} \) |
| 37 | \( 1 + 7.12T + 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 + 5.43T + 43T^{2} \) |
| 47 | \( 1 + 3.68T + 47T^{2} \) |
| 53 | \( 1 - 4.43T + 53T^{2} \) |
| 59 | \( 1 - 4.43T + 59T^{2} \) |
| 61 | \( 1 + 14.8T + 61T^{2} \) |
| 67 | \( 1 - 0.438T + 67T^{2} \) |
| 71 | \( 1 + 4.24T + 71T^{2} \) |
| 73 | \( 1 + 9.24T + 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - 17.3T + 83T^{2} \) |
| 89 | \( 1 + 15.3T + 89T^{2} \) |
| 97 | \( 1 - 7.12T + 97T^{2} \) |
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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
−9.575906939164834266073419789204, −8.843029389839722896636014799722, −8.290837099466879740148326930035, −7.45600856960558964340236169379, −6.33829063977038368175117962003, −5.47284661848185631628299995762, −4.66217418734000829038686331613, −3.34944380474305733188639208898, −2.35216058809248867116473619110, −1.48887825491620288011874390771,
1.48887825491620288011874390771, 2.35216058809248867116473619110, 3.34944380474305733188639208898, 4.66217418734000829038686331613, 5.47284661848185631628299995762, 6.33829063977038368175117962003, 7.45600856960558964340236169379, 8.290837099466879740148326930035, 8.843029389839722896636014799722, 9.575906939164834266073419789204