| L(s) = 1 | + 1.78·2-s + 1.19·4-s + 0.326·5-s + 1.17·7-s − 1.44·8-s + 0.582·10-s − 0.852·11-s + 3.22·13-s + 2.09·14-s − 4.96·16-s − 5.35·17-s − 1.49·19-s + 0.388·20-s − 1.52·22-s − 23-s − 4.89·25-s + 5.76·26-s + 1.39·28-s − 29-s + 0.352·31-s − 5.97·32-s − 9.57·34-s + 0.382·35-s − 10.9·37-s − 2.66·38-s − 0.471·40-s + 11.6·41-s + ⋯ |
| L(s) = 1 | + 1.26·2-s + 0.595·4-s + 0.145·5-s + 0.443·7-s − 0.510·8-s + 0.184·10-s − 0.256·11-s + 0.894·13-s + 0.559·14-s − 1.24·16-s − 1.29·17-s − 0.342·19-s + 0.0868·20-s − 0.324·22-s − 0.208·23-s − 0.978·25-s + 1.13·26-s + 0.263·28-s − 0.185·29-s + 0.0633·31-s − 1.05·32-s − 1.64·34-s + 0.0646·35-s − 1.79·37-s − 0.432·38-s − 0.0745·40-s + 1.82·41-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 | \( 1 - 1.78T + 2T^{2} \) |
| 5 | \( 1 - 0.326T + 5T^{2} \) |
| 7 | \( 1 - 1.17T + 7T^{2} \) |
| 11 | \( 1 + 0.852T + 11T^{2} \) |
| 13 | \( 1 - 3.22T + 13T^{2} \) |
| 17 | \( 1 + 5.35T + 17T^{2} \) |
| 19 | \( 1 + 1.49T + 19T^{2} \) |
| 31 | \( 1 - 0.352T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 + 3.16T + 43T^{2} \) |
| 47 | \( 1 + 1.44T + 47T^{2} \) |
| 53 | \( 1 + 6.05T + 53T^{2} \) |
| 59 | \( 1 + 1.56T + 59T^{2} \) |
| 61 | \( 1 + 6.48T + 61T^{2} \) |
| 67 | \( 1 + 6.67T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 + 15.9T + 79T^{2} \) |
| 83 | \( 1 - 4.84T + 83T^{2} \) |
| 89 | \( 1 - 7.47T + 89T^{2} \) |
| 97 | \( 1 - 3.68T + 97T^{2} \) |
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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
−7.66395698707775141126467392134, −6.64328302302450153635411761573, −6.21779531027237977658088086926, −5.46054517519678188823946442286, −4.78463661544474751810129456928, −4.11232706667150952393884015752, −3.48306286285064452704315619340, −2.48908561407048795063155224188, −1.68324109782085918965266447839, 0,
1.68324109782085918965266447839, 2.48908561407048795063155224188, 3.48306286285064452704315619340, 4.11232706667150952393884015752, 4.78463661544474751810129456928, 5.46054517519678188823946442286, 6.21779531027237977658088086926, 6.64328302302450153635411761573, 7.66395698707775141126467392134
