| L(s) = 1 | − 2-s − 1.28·3-s + 4-s − 1.76·5-s + 1.28·6-s − 7-s − 8-s − 1.35·9-s + 1.76·10-s + 2.36·11-s − 1.28·12-s + 4.94·13-s + 14-s + 2.26·15-s + 16-s − 5.35·17-s + 1.35·18-s − 3.39·19-s − 1.76·20-s + 1.28·21-s − 2.36·22-s − 1.24·23-s + 1.28·24-s − 1.88·25-s − 4.94·26-s + 5.58·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.740·3-s + 0.5·4-s − 0.789·5-s + 0.523·6-s − 0.377·7-s − 0.353·8-s − 0.451·9-s + 0.558·10-s + 0.713·11-s − 0.370·12-s + 1.37·13-s + 0.267·14-s + 0.584·15-s + 0.250·16-s − 1.29·17-s + 0.319·18-s − 0.779·19-s − 0.394·20-s + 0.279·21-s − 0.504·22-s − 0.259·23-s + 0.261·24-s − 0.377·25-s − 0.968·26-s + 1.07·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 + 1.28T + 3T^{2} \) |
| 5 | \( 1 + 1.76T + 5T^{2} \) |
| 11 | \( 1 - 2.36T + 11T^{2} \) |
| 13 | \( 1 - 4.94T + 13T^{2} \) |
| 17 | \( 1 + 5.35T + 17T^{2} \) |
| 19 | \( 1 + 3.39T + 19T^{2} \) |
| 23 | \( 1 + 1.24T + 23T^{2} \) |
| 29 | \( 1 - 3.78T + 29T^{2} \) |
| 31 | \( 1 - 7.86T + 31T^{2} \) |
| 37 | \( 1 + 2.11T + 37T^{2} \) |
| 41 | \( 1 + 6.47T + 41T^{2} \) |
| 43 | \( 1 + 3.36T + 43T^{2} \) |
| 47 | \( 1 - 7.31T + 47T^{2} \) |
| 53 | \( 1 + 6.54T + 53T^{2} \) |
| 59 | \( 1 + 4.49T + 59T^{2} \) |
| 61 | \( 1 + 1.67T + 61T^{2} \) |
| 67 | \( 1 - 15.0T + 67T^{2} \) |
| 71 | \( 1 - 0.648T + 71T^{2} \) |
| 73 | \( 1 - 15.2T + 73T^{2} \) |
| 79 | \( 1 + 1.47T + 79T^{2} \) |
| 83 | \( 1 + 0.837T + 83T^{2} \) |
| 89 | \( 1 + 0.643T + 89T^{2} \) |
| 97 | \( 1 + 16.5T + 97T^{2} \) |
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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
−7.918988493169234442141561349239, −6.68809855078153031713941151738, −6.55124344148422491231255334424, −5.89704566720155306572871095857, −4.78847979209173198819094412832, −4.01244215215265577520377879741, −3.27094593572216592493981293016, −2.16135463785623762136797489672, −0.953721112377440778733946792862, 0,
0.953721112377440778733946792862, 2.16135463785623762136797489672, 3.27094593572216592493981293016, 4.01244215215265577520377879741, 4.78847979209173198819094412832, 5.89704566720155306572871095857, 6.55124344148422491231255334424, 6.68809855078153031713941151738, 7.918988493169234442141561349239
