| L(s) = 1 | − 2-s − 1.83·3-s + 4-s − 5-s + 1.83·6-s + 1.88·7-s − 8-s + 0.357·9-s + 10-s + 1.98·11-s − 1.83·12-s − 4.14·13-s − 1.88·14-s + 1.83·15-s + 16-s − 4.84·17-s − 0.357·18-s + 5.13·19-s − 20-s − 3.45·21-s − 1.98·22-s − 6.83·23-s + 1.83·24-s + 25-s + 4.14·26-s + 4.84·27-s + 1.88·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.05·3-s + 0.5·4-s − 0.447·5-s + 0.748·6-s + 0.711·7-s − 0.353·8-s + 0.119·9-s + 0.316·10-s + 0.598·11-s − 0.528·12-s − 1.15·13-s − 0.503·14-s + 0.473·15-s + 0.250·16-s − 1.17·17-s − 0.0843·18-s + 1.17·19-s − 0.223·20-s − 0.752·21-s − 0.423·22-s − 1.42·23-s + 0.374·24-s + 0.200·25-s + 0.813·26-s + 0.931·27-s + 0.355·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 601 | \( 1 + T \) |
| good | 3 | \( 1 + 1.83T + 3T^{2} \) |
| 7 | \( 1 - 1.88T + 7T^{2} \) |
| 11 | \( 1 - 1.98T + 11T^{2} \) |
| 13 | \( 1 + 4.14T + 13T^{2} \) |
| 17 | \( 1 + 4.84T + 17T^{2} \) |
| 19 | \( 1 - 5.13T + 19T^{2} \) |
| 23 | \( 1 + 6.83T + 23T^{2} \) |
| 29 | \( 1 + 1.09T + 29T^{2} \) |
| 31 | \( 1 - 10.0T + 31T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 + 1.07T + 41T^{2} \) |
| 43 | \( 1 + 1.33T + 43T^{2} \) |
| 47 | \( 1 + 3.28T + 47T^{2} \) |
| 53 | \( 1 + 8.56T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 + 9.29T + 61T^{2} \) |
| 67 | \( 1 + 2.41T + 67T^{2} \) |
| 71 | \( 1 - 6.07T + 71T^{2} \) |
| 73 | \( 1 - 0.125T + 73T^{2} \) |
| 79 | \( 1 + 7.97T + 79T^{2} \) |
| 83 | \( 1 + 3.05T + 83T^{2} \) |
| 89 | \( 1 - 4.03T + 89T^{2} \) |
| 97 | \( 1 - 6.65T + 97T^{2} \) |
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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
−7.894424016367700622891766695293, −6.98445388118877722463104663619, −6.40044176357889016063823405079, −5.71454032860242011607710418321, −4.75774938888437341556587169327, −4.38985273318789232008236651062, −3.06306437548411566037101573209, −2.11723040474680074358547249928, −1.00519958814402914854611534369, 0,
1.00519958814402914854611534369, 2.11723040474680074358547249928, 3.06306437548411566037101573209, 4.38985273318789232008236651062, 4.75774938888437341556587169327, 5.71454032860242011607710418321, 6.40044176357889016063823405079, 6.98445388118877722463104663619, 7.894424016367700622891766695293
