| L(s) = 1 | + 5·3-s + 5·5-s + 20.3·7-s − 2·9-s − 61.1·13-s + 25·15-s + 20.3·17-s − 101.·19-s + 101.·21-s − 35·23-s − 100·25-s − 145·27-s + 203.·29-s − 15·31-s + 101.·35-s − 265·37-s − 305.·39-s + 101.·41-s − 448.·43-s − 10·45-s − 380·47-s + 72.9·49-s + 101.·51-s + 510·53-s − 509.·57-s − 21·59-s − 203.·61-s + ⋯ |
| L(s) = 1 | + 0.962·3-s + 0.447·5-s + 1.10·7-s − 0.0740·9-s − 1.30·13-s + 0.430·15-s + 0.290·17-s − 1.23·19-s + 1.05·21-s − 0.317·23-s − 0.800·25-s − 1.03·27-s + 1.30·29-s − 0.0869·31-s + 0.492·35-s − 1.17·37-s − 1.25·39-s + 0.388·41-s − 1.59·43-s − 0.0331·45-s − 1.17·47-s + 0.212·49-s + 0.280·51-s + 1.32·53-s − 1.18·57-s − 0.0463·59-s − 0.428·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1936 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 5T + 27T^{2} \) |
| 5 | \( 1 - 5T + 125T^{2} \) |
| 7 | \( 1 - 20.3T + 343T^{2} \) |
| 13 | \( 1 + 61.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 20.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 101.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 35T + 1.21e4T^{2} \) |
| 29 | \( 1 - 203.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 15T + 2.97e4T^{2} \) |
| 37 | \( 1 + 265T + 5.06e4T^{2} \) |
| 41 | \( 1 - 101.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 448.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 380T + 1.03e5T^{2} \) |
| 53 | \( 1 - 510T + 1.48e5T^{2} \) |
| 59 | \( 1 + 21T + 2.05e5T^{2} \) |
| 61 | \( 1 + 203.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 585T + 3.00e5T^{2} \) |
| 71 | \( 1 + 313T + 3.57e5T^{2} \) |
| 73 | \( 1 + 469.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 611.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 652.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 185T + 7.04e5T^{2} \) |
| 97 | \( 1 - 785T + 9.12e5T^{2} \) |
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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
−8.406238717581870970665467471004, −7.892567963769392815710824141348, −7.05282752871815987031633963380, −6.02356267396606478462323700842, −5.07533305330982728582963241080, −4.38732673657760975024727987202, −3.23045038660188913462276342097, −2.27377461151570966139577951310, −1.69519789516285167485933799656, 0,
1.69519789516285167485933799656, 2.27377461151570966139577951310, 3.23045038660188913462276342097, 4.38732673657760975024727987202, 5.07533305330982728582963241080, 6.02356267396606478462323700842, 7.05282752871815987031633963380, 7.892567963769392815710824141348, 8.406238717581870970665467471004
