| L(s) = 1 | + 2-s − 3-s + 4-s + 0.717·5-s − 6-s + 7-s + 8-s + 9-s + 0.717·10-s − 6.48·11-s − 12-s + 13-s + 14-s − 0.717·15-s + 16-s + 17-s + 18-s + 4.03·19-s + 0.717·20-s − 21-s − 6.48·22-s − 2.21·23-s − 24-s − 4.48·25-s + 26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.320·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.226·10-s − 1.95·11-s − 0.288·12-s + 0.277·13-s + 0.267·14-s − 0.185·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s + 0.925·19-s + 0.160·20-s − 0.218·21-s − 1.38·22-s − 0.462·23-s − 0.204·24-s − 0.896·25-s + 0.196·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9282 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9282 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 0.717T + 5T^{2} \) |
| 11 | \( 1 + 6.48T + 11T^{2} \) |
| 19 | \( 1 - 4.03T + 19T^{2} \) |
| 23 | \( 1 + 2.21T + 23T^{2} \) |
| 29 | \( 1 + 5.86T + 29T^{2} \) |
| 31 | \( 1 - 7.23T + 31T^{2} \) |
| 37 | \( 1 - 1.59T + 37T^{2} \) |
| 41 | \( 1 + 9.50T + 41T^{2} \) |
| 43 | \( 1 + 2.34T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 + 11.5T + 53T^{2} \) |
| 59 | \( 1 + 3.41T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 + 3.67T + 67T^{2} \) |
| 71 | \( 1 + 1.04T + 71T^{2} \) |
| 73 | \( 1 + 2.74T + 73T^{2} \) |
| 79 | \( 1 + 3.50T + 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 - 8.63T + 89T^{2} \) |
| 97 | \( 1 - 2.60T + 97T^{2} \) |
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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
−7.41636043642167416023136866721, −6.51490739576341650177367271553, −5.69346372076459069141702234460, −5.42154577777689619873265375886, −4.78493069746728407464104599396, −3.94821126755601562941703962448, −3.03145896568350244996745099537, −2.29903088788854665603286551136, −1.36693167608977323589753744023, 0,
1.36693167608977323589753744023, 2.29903088788854665603286551136, 3.03145896568350244996745099537, 3.94821126755601562941703962448, 4.78493069746728407464104599396, 5.42154577777689619873265375886, 5.69346372076459069141702234460, 6.51490739576341650177367271553, 7.41636043642167416023136866721
