L(s) = 1 | + 3-s + 1.02·5-s + 1.14·7-s + 9-s + 1.84·11-s + 3.42·13-s + 1.02·15-s + 4.07·17-s + 7.18·19-s + 1.14·21-s − 2.14·23-s − 3.95·25-s + 27-s − 2.76·29-s + 8.05·31-s + 1.84·33-s + 1.16·35-s + 3.19·37-s + 3.42·39-s + 7.47·41-s − 9.54·43-s + 1.02·45-s + 3.93·47-s − 5.68·49-s + 4.07·51-s − 4.47·53-s + 1.88·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.456·5-s + 0.432·7-s + 0.333·9-s + 0.556·11-s + 0.951·13-s + 0.263·15-s + 0.987·17-s + 1.64·19-s + 0.249·21-s − 0.447·23-s − 0.791·25-s + 0.192·27-s − 0.513·29-s + 1.44·31-s + 0.321·33-s + 0.197·35-s + 0.525·37-s + 0.549·39-s + 1.16·41-s − 1.45·43-s + 0.152·45-s + 0.574·47-s − 0.812·49-s + 0.570·51-s − 0.614·53-s + 0.254·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.798290060\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.798290060\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 + T \) |
good | 5 | \( 1 - 1.02T + 5T^{2} \) |
| 7 | \( 1 - 1.14T + 7T^{2} \) |
| 11 | \( 1 - 1.84T + 11T^{2} \) |
| 13 | \( 1 - 3.42T + 13T^{2} \) |
| 17 | \( 1 - 4.07T + 17T^{2} \) |
| 19 | \( 1 - 7.18T + 19T^{2} \) |
| 23 | \( 1 + 2.14T + 23T^{2} \) |
| 29 | \( 1 + 2.76T + 29T^{2} \) |
| 31 | \( 1 - 8.05T + 31T^{2} \) |
| 37 | \( 1 - 3.19T + 37T^{2} \) |
| 41 | \( 1 - 7.47T + 41T^{2} \) |
| 43 | \( 1 + 9.54T + 43T^{2} \) |
| 47 | \( 1 - 3.93T + 47T^{2} \) |
| 53 | \( 1 + 4.47T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 + 0.884T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 - 0.259T + 71T^{2} \) |
| 73 | \( 1 + 5.24T + 73T^{2} \) |
| 79 | \( 1 - 8.10T + 79T^{2} \) |
| 83 | \( 1 + 3.31T + 83T^{2} \) |
| 89 | \( 1 + 1.15T + 89T^{2} \) |
| 97 | \( 1 + 11.1T + 97T^{2} \) |
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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
−7.890317267076518558590713674639, −7.31730397916718025825762735944, −6.35845230697293764535495178754, −5.81235449735233506439317362045, −5.08016492415512329087956149492, −4.15139324249806930670201526873, −3.48085849418744564652982506759, −2.74402105302633410788174500560, −1.64279563175781847844154410871, −1.05789013382280532815839307908,
1.05789013382280532815839307908, 1.64279563175781847844154410871, 2.74402105302633410788174500560, 3.48085849418744564652982506759, 4.15139324249806930670201526873, 5.08016492415512329087956149492, 5.81235449735233506439317362045, 6.35845230697293764535495178754, 7.31730397916718025825762735944, 7.890317267076518558590713674639