| L(s) = 1 | − 2.68·2-s + 5.22·4-s − 0.687·5-s − 3.46·7-s − 8.66·8-s + 1.84·10-s + 4.59·11-s − 5.14·13-s + 9.30·14-s + 12.8·16-s + 5.97·17-s + 2.27·19-s − 3.59·20-s − 12.3·22-s + 23-s − 4.52·25-s + 13.8·26-s − 18.0·28-s + 29-s + 1.71·31-s − 17.1·32-s − 16.0·34-s + 2.38·35-s + 1.76·37-s − 6.12·38-s + 5.95·40-s + 0.217·41-s + ⋯ |
| L(s) = 1 | − 1.90·2-s + 2.61·4-s − 0.307·5-s − 1.30·7-s − 3.06·8-s + 0.584·10-s + 1.38·11-s − 1.42·13-s + 2.48·14-s + 3.21·16-s + 1.44·17-s + 0.523·19-s − 0.803·20-s − 2.63·22-s + 0.208·23-s − 0.905·25-s + 2.71·26-s − 3.41·28-s + 0.185·29-s + 0.308·31-s − 3.03·32-s − 2.75·34-s + 0.402·35-s + 0.289·37-s − 0.994·38-s + 0.942·40-s + 0.0339·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 2.68T + 2T^{2} \) |
| 5 | \( 1 + 0.687T + 5T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 - 4.59T + 11T^{2} \) |
| 13 | \( 1 + 5.14T + 13T^{2} \) |
| 17 | \( 1 - 5.97T + 17T^{2} \) |
| 19 | \( 1 - 2.27T + 19T^{2} \) |
| 31 | \( 1 - 1.71T + 31T^{2} \) |
| 37 | \( 1 - 1.76T + 37T^{2} \) |
| 41 | \( 1 - 0.217T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 + 3.23T + 47T^{2} \) |
| 53 | \( 1 + 9.45T + 53T^{2} \) |
| 59 | \( 1 + 4.74T + 59T^{2} \) |
| 61 | \( 1 - 0.257T + 61T^{2} \) |
| 67 | \( 1 + 12.3T + 67T^{2} \) |
| 71 | \( 1 - 4.10T + 71T^{2} \) |
| 73 | \( 1 - 3.50T + 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 - 5.50T + 83T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 - 18.9T + 97T^{2} \) |
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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
−7.61689319890910860045118920675, −7.40858066279374112402330307374, −6.41767204234274100787708147243, −6.22474849543249464440016630215, −4.97290703172238463320706493325, −3.52179565131606956412439701294, −3.11130040070652379805163996801, −2.00119601509055529292394868354, −0.998680315493724787531339416015, 0,
0.998680315493724787531339416015, 2.00119601509055529292394868354, 3.11130040070652379805163996801, 3.52179565131606956412439701294, 4.97290703172238463320706493325, 6.22474849543249464440016630215, 6.41767204234274100787708147243, 7.40858066279374112402330307374, 7.61689319890910860045118920675
