| L(s) = 1 | − 2-s + 3-s + 4-s + 3.49·5-s − 6-s − 4.54·7-s − 8-s + 9-s − 3.49·10-s − 4.43·11-s + 12-s − 0.549·13-s + 4.54·14-s + 3.49·15-s + 16-s − 17-s − 18-s + 7.86·19-s + 3.49·20-s − 4.54·21-s + 4.43·22-s − 0.106·23-s − 24-s + 7.22·25-s + 0.549·26-s + 27-s − 4.54·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.56·5-s − 0.408·6-s − 1.71·7-s − 0.353·8-s + 0.333·9-s − 1.10·10-s − 1.33·11-s + 0.288·12-s − 0.152·13-s + 1.21·14-s + 0.902·15-s + 0.250·16-s − 0.242·17-s − 0.235·18-s + 1.80·19-s + 0.781·20-s − 0.991·21-s + 0.946·22-s − 0.0222·23-s − 0.204·24-s + 1.44·25-s + 0.107·26-s + 0.192·27-s − 0.858·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 5 | \( 1 - 3.49T + 5T^{2} \) |
| 7 | \( 1 + 4.54T + 7T^{2} \) |
| 11 | \( 1 + 4.43T + 11T^{2} \) |
| 13 | \( 1 + 0.549T + 13T^{2} \) |
| 19 | \( 1 - 7.86T + 19T^{2} \) |
| 23 | \( 1 + 0.106T + 23T^{2} \) |
| 29 | \( 1 + 9.87T + 29T^{2} \) |
| 31 | \( 1 - 2.98T + 31T^{2} \) |
| 37 | \( 1 + 6.87T + 37T^{2} \) |
| 41 | \( 1 - 7.13T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 - 6.14T + 47T^{2} \) |
| 53 | \( 1 - 13.0T + 53T^{2} \) |
| 61 | \( 1 + 6.91T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 - 4.03T + 73T^{2} \) |
| 79 | \( 1 + 2.01T + 79T^{2} \) |
| 83 | \( 1 + 3.43T + 83T^{2} \) |
| 89 | \( 1 + 0.242T + 89T^{2} \) |
| 97 | \( 1 - 0.142T + 97T^{2} \) |
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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
−7.52132354461938235276474142830, −7.25001821183961332120532901521, −6.32926947636319323442859814260, −5.69648047952875602075742921431, −5.18754610455701777735652428379, −3.66887639564890199354881260973, −2.86544969827018813630414342922, −2.45827171803306422653030980830, −1.40142390594414191871508019827, 0,
1.40142390594414191871508019827, 2.45827171803306422653030980830, 2.86544969827018813630414342922, 3.66887639564890199354881260973, 5.18754610455701777735652428379, 5.69648047952875602075742921431, 6.32926947636319323442859814260, 7.25001821183961332120532901521, 7.52132354461938235276474142830
