L(s) = 1 | + 3.42·3-s − 2.35·5-s + 8.72·9-s − 0.142·11-s + 2.89·13-s − 8.07·15-s + 1.39·17-s + 7.23·19-s − 4.70·23-s + 0.557·25-s + 19.5·27-s + 2.50·29-s + 2.29·31-s − 0.489·33-s − 8.95·37-s + 9.89·39-s + 41-s + 10.0·43-s − 20.5·45-s + 0.577·47-s + 4.77·51-s − 2.67·53-s + 0.336·55-s + 24.7·57-s + 5.00·59-s − 10.1·61-s − 6.81·65-s + ⋯ |
L(s) = 1 | + 1.97·3-s − 1.05·5-s + 2.90·9-s − 0.0430·11-s + 0.801·13-s − 2.08·15-s + 0.338·17-s + 1.65·19-s − 0.980·23-s + 0.111·25-s + 3.77·27-s + 0.464·29-s + 0.412·31-s − 0.0851·33-s − 1.47·37-s + 1.58·39-s + 0.156·41-s + 1.53·43-s − 3.06·45-s + 0.0842·47-s + 0.668·51-s − 0.367·53-s + 0.0454·55-s + 3.28·57-s + 0.651·59-s − 1.30·61-s − 0.845·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.303133923\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.303133923\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
good | 3 | \( 1 - 3.42T + 3T^{2} \) |
| 5 | \( 1 + 2.35T + 5T^{2} \) |
| 11 | \( 1 + 0.142T + 11T^{2} \) |
| 13 | \( 1 - 2.89T + 13T^{2} \) |
| 17 | \( 1 - 1.39T + 17T^{2} \) |
| 19 | \( 1 - 7.23T + 19T^{2} \) |
| 23 | \( 1 + 4.70T + 23T^{2} \) |
| 29 | \( 1 - 2.50T + 29T^{2} \) |
| 31 | \( 1 - 2.29T + 31T^{2} \) |
| 37 | \( 1 + 8.95T + 37T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 - 0.577T + 47T^{2} \) |
| 53 | \( 1 + 2.67T + 53T^{2} \) |
| 59 | \( 1 - 5.00T + 59T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 - 4.81T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 - 2.06T + 73T^{2} \) |
| 79 | \( 1 - 2.74T + 79T^{2} \) |
| 83 | \( 1 + 8.58T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 - 4.73T + 97T^{2} \) |
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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
−7.70648099264475024416879986977, −7.60759580728992157395131901756, −6.82189880483440455979642270017, −5.76676613835230904611520890569, −4.64213078458882162332042250489, −4.01071832529982475373461283005, −3.41112472002290313618911201514, −2.95160316318605092613959507390, −1.89790108632184992450301431597, −0.987735427887603126629375657551,
0.987735427887603126629375657551, 1.89790108632184992450301431597, 2.95160316318605092613959507390, 3.41112472002290313618911201514, 4.01071832529982475373461283005, 4.64213078458882162332042250489, 5.76676613835230904611520890569, 6.82189880483440455979642270017, 7.60759580728992157395131901756, 7.70648099264475024416879986977