| L(s) = 1 | + 0.806·2-s − 1.34·4-s − 2.04·5-s + 4.05·7-s − 2.70·8-s − 1.64·10-s + 3.00·11-s + 1.84·13-s + 3.27·14-s + 0.519·16-s − 7.58·17-s − 2.21·19-s + 2.75·20-s + 2.42·22-s + 23-s − 0.817·25-s + 1.48·26-s − 5.47·28-s − 29-s + 2.91·31-s + 5.82·32-s − 6.11·34-s − 8.29·35-s − 4.19·37-s − 1.79·38-s + 5.52·40-s − 7.56·41-s + ⋯ |
| L(s) = 1 | + 0.570·2-s − 0.674·4-s − 0.914·5-s + 1.53·7-s − 0.955·8-s − 0.521·10-s + 0.905·11-s + 0.511·13-s + 0.873·14-s + 0.129·16-s − 1.83·17-s − 0.509·19-s + 0.617·20-s + 0.516·22-s + 0.208·23-s − 0.163·25-s + 0.291·26-s − 1.03·28-s − 0.185·29-s + 0.524·31-s + 1.02·32-s − 1.04·34-s − 1.40·35-s − 0.689·37-s − 0.290·38-s + 0.873·40-s − 1.18·41-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 | \( 1 - 0.806T + 2T^{2} \) |
| 5 | \( 1 + 2.04T + 5T^{2} \) |
| 7 | \( 1 - 4.05T + 7T^{2} \) |
| 11 | \( 1 - 3.00T + 11T^{2} \) |
| 13 | \( 1 - 1.84T + 13T^{2} \) |
| 17 | \( 1 + 7.58T + 17T^{2} \) |
| 19 | \( 1 + 2.21T + 19T^{2} \) |
| 31 | \( 1 - 2.91T + 31T^{2} \) |
| 37 | \( 1 + 4.19T + 37T^{2} \) |
| 41 | \( 1 + 7.56T + 41T^{2} \) |
| 43 | \( 1 - 1.21T + 43T^{2} \) |
| 47 | \( 1 + 2.01T + 47T^{2} \) |
| 53 | \( 1 + 4.03T + 53T^{2} \) |
| 59 | \( 1 + 4.61T + 59T^{2} \) |
| 61 | \( 1 - 6.09T + 61T^{2} \) |
| 67 | \( 1 - 3.87T + 67T^{2} \) |
| 71 | \( 1 - 14.7T + 71T^{2} \) |
| 73 | \( 1 - 6.28T + 73T^{2} \) |
| 79 | \( 1 - 8.58T + 79T^{2} \) |
| 83 | \( 1 - 2.36T + 83T^{2} \) |
| 89 | \( 1 - 9.84T + 89T^{2} \) |
| 97 | \( 1 + 16.9T + 97T^{2} \) |
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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
−8.050928774515035090828828735909, −6.85896970926537392733244939435, −6.37377793376737832769817945323, −5.22740361753942451034272418775, −4.76342201948362541576197911674, −4.03021410550507640855016467137, −3.72090510229933237057657747196, −2.35892797769243063917746915064, −1.32497510583344511135496998216, 0,
1.32497510583344511135496998216, 2.35892797769243063917746915064, 3.72090510229933237057657747196, 4.03021410550507640855016467137, 4.76342201948362541576197911674, 5.22740361753942451034272418775, 6.37377793376737832769817945323, 6.85896970926537392733244939435, 8.050928774515035090828828735909
