L(s) = 1 | + 2.82·5-s + 4.85·7-s − 3·9-s + 1.69·11-s + 3.33·13-s − 2.21·17-s + 19-s + 5.35·23-s + 2.96·25-s − 6.61·29-s + 1.16·31-s + 13.7·35-s + 4.72·37-s − 3.45·41-s − 3.96·43-s − 8.46·45-s + 0.312·47-s + 16.6·49-s + 5.80·53-s + 4.77·55-s + 6.38·59-s + 3.99·61-s − 14.5·63-s + 9.42·65-s − 1.03·67-s + 15.9·71-s − 16.0·73-s + ⋯ |
L(s) = 1 | + 1.26·5-s + 1.83·7-s − 9-s + 0.510·11-s + 0.926·13-s − 0.536·17-s + 0.229·19-s + 1.11·23-s + 0.592·25-s − 1.22·29-s + 0.208·31-s + 2.31·35-s + 0.777·37-s − 0.540·41-s − 0.605·43-s − 1.26·45-s + 0.0456·47-s + 2.37·49-s + 0.797·53-s + 0.644·55-s + 0.831·59-s + 0.511·61-s − 1.83·63-s + 1.16·65-s − 0.126·67-s + 1.89·71-s − 1.87·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.447752869\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.447752869\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| 79 | \( 1 - T \) |
good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 - 2.82T + 5T^{2} \) |
| 7 | \( 1 - 4.85T + 7T^{2} \) |
| 11 | \( 1 - 1.69T + 11T^{2} \) |
| 13 | \( 1 - 3.33T + 13T^{2} \) |
| 17 | \( 1 + 2.21T + 17T^{2} \) |
| 23 | \( 1 - 5.35T + 23T^{2} \) |
| 29 | \( 1 + 6.61T + 29T^{2} \) |
| 31 | \( 1 - 1.16T + 31T^{2} \) |
| 37 | \( 1 - 4.72T + 37T^{2} \) |
| 41 | \( 1 + 3.45T + 41T^{2} \) |
| 43 | \( 1 + 3.96T + 43T^{2} \) |
| 47 | \( 1 - 0.312T + 47T^{2} \) |
| 53 | \( 1 - 5.80T + 53T^{2} \) |
| 59 | \( 1 - 6.38T + 59T^{2} \) |
| 61 | \( 1 - 3.99T + 61T^{2} \) |
| 67 | \( 1 + 1.03T + 67T^{2} \) |
| 71 | \( 1 - 15.9T + 71T^{2} \) |
| 73 | \( 1 + 16.0T + 73T^{2} \) |
| 83 | \( 1 + 1.04T + 83T^{2} \) |
| 89 | \( 1 + 16.2T + 89T^{2} \) |
| 97 | \( 1 + 6.30T + 97T^{2} \) |
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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
−8.303274379251654943750285660033, −7.39142540645293229808107911355, −6.56958969100160247023131840551, −5.74565643251285217405851440482, −5.37721340914705632393773713355, −4.61772912687028466078159975575, −3.66475173694483580133822161583, −2.53632258382607028270638653778, −1.82056650467146323526691565254, −1.07198056032298388612842116247,
1.07198056032298388612842116247, 1.82056650467146323526691565254, 2.53632258382607028270638653778, 3.66475173694483580133822161583, 4.61772912687028466078159975575, 5.37721340914705632393773713355, 5.74565643251285217405851440482, 6.56958969100160247023131840551, 7.39142540645293229808107911355, 8.303274379251654943750285660033