| L(s) = 1 | + 2.75·2-s + 5.57·4-s − 5-s − 4.05·7-s + 9.82·8-s − 2.75·10-s − 5.98·11-s − 1.19·13-s − 11.1·14-s + 15.8·16-s − 4.77·17-s − 4.76·19-s − 5.57·20-s − 16.4·22-s + 0.291·23-s + 25-s − 3.27·26-s − 22.5·28-s + 4.91·29-s − 2.46·31-s + 24.0·32-s − 13.1·34-s + 4.05·35-s + 4.09·37-s − 13.1·38-s − 9.82·40-s + 5.80·41-s + ⋯ |
| L(s) = 1 | + 1.94·2-s + 2.78·4-s − 0.447·5-s − 1.53·7-s + 3.47·8-s − 0.870·10-s − 1.80·11-s − 0.330·13-s − 2.98·14-s + 3.97·16-s − 1.15·17-s − 1.09·19-s − 1.24·20-s − 3.50·22-s + 0.0606·23-s + 0.200·25-s − 0.643·26-s − 4.27·28-s + 0.913·29-s − 0.442·31-s + 4.25·32-s − 2.25·34-s + 0.685·35-s + 0.673·37-s − 2.12·38-s − 1.55·40-s + 0.906·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 89 | \( 1 + T \) |
| good | 2 | \( 1 - 2.75T + 2T^{2} \) |
| 7 | \( 1 + 4.05T + 7T^{2} \) |
| 11 | \( 1 + 5.98T + 11T^{2} \) |
| 13 | \( 1 + 1.19T + 13T^{2} \) |
| 17 | \( 1 + 4.77T + 17T^{2} \) |
| 19 | \( 1 + 4.76T + 19T^{2} \) |
| 23 | \( 1 - 0.291T + 23T^{2} \) |
| 29 | \( 1 - 4.91T + 29T^{2} \) |
| 31 | \( 1 + 2.46T + 31T^{2} \) |
| 37 | \( 1 - 4.09T + 37T^{2} \) |
| 41 | \( 1 - 5.80T + 41T^{2} \) |
| 43 | \( 1 + 8.92T + 43T^{2} \) |
| 47 | \( 1 - 3.80T + 47T^{2} \) |
| 53 | \( 1 + 9.51T + 53T^{2} \) |
| 59 | \( 1 + 9.57T + 59T^{2} \) |
| 61 | \( 1 + 5.87T + 61T^{2} \) |
| 67 | \( 1 + 4.40T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 + 5.72T + 73T^{2} \) |
| 79 | \( 1 + 6.02T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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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
−7.64237532040682909442949828188, −7.11784358114645181019882757523, −6.24991893537317809957805557933, −5.96244730689978794108108327954, −4.77158621041925909627254341512, −4.48937528286469101529804865804, −3.36432991691746045983364987345, −2.85459083406091466771062545038, −2.15907210837115267882391365778, 0,
2.15907210837115267882391365778, 2.85459083406091466771062545038, 3.36432991691746045983364987345, 4.48937528286469101529804865804, 4.77158621041925909627254341512, 5.96244730689978794108108327954, 6.24991893537317809957805557933, 7.11784358114645181019882757523, 7.64237532040682909442949828188
