| L(s) = 1 | − 1.25·2-s + 3-s − 0.417·4-s + 5-s − 1.25·6-s − 1.35·7-s + 3.04·8-s + 9-s − 1.25·10-s − 0.345·11-s − 0.417·12-s − 13-s + 1.70·14-s + 15-s − 2.99·16-s − 6.69·17-s − 1.25·18-s + 1.57·19-s − 0.417·20-s − 1.35·21-s + 0.434·22-s + 5.40·23-s + 3.04·24-s + 25-s + 1.25·26-s + 27-s + 0.566·28-s + ⋯ |
| L(s) = 1 | − 0.889·2-s + 0.577·3-s − 0.208·4-s + 0.447·5-s − 0.513·6-s − 0.513·7-s + 1.07·8-s + 0.333·9-s − 0.397·10-s − 0.104·11-s − 0.120·12-s − 0.277·13-s + 0.456·14-s + 0.258·15-s − 0.747·16-s − 1.62·17-s − 0.296·18-s + 0.361·19-s − 0.0932·20-s − 0.296·21-s + 0.0925·22-s + 1.12·23-s + 0.620·24-s + 0.200·25-s + 0.246·26-s + 0.192·27-s + 0.107·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 1.25T + 2T^{2} \) |
| 7 | \( 1 + 1.35T + 7T^{2} \) |
| 11 | \( 1 + 0.345T + 11T^{2} \) |
| 17 | \( 1 + 6.69T + 17T^{2} \) |
| 19 | \( 1 - 1.57T + 19T^{2} \) |
| 23 | \( 1 - 5.40T + 23T^{2} \) |
| 29 | \( 1 - 0.119T + 29T^{2} \) |
| 37 | \( 1 + 3.44T + 37T^{2} \) |
| 41 | \( 1 - 1.58T + 41T^{2} \) |
| 43 | \( 1 - 7.03T + 43T^{2} \) |
| 47 | \( 1 - 2.35T + 47T^{2} \) |
| 53 | \( 1 + 2.33T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 - 1.27T + 67T^{2} \) |
| 71 | \( 1 - 2.26T + 71T^{2} \) |
| 73 | \( 1 + 9.90T + 73T^{2} \) |
| 79 | \( 1 - 3.65T + 79T^{2} \) |
| 83 | \( 1 + 11.9T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 + 15.1T + 97T^{2} \) |
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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
−7.82891765194818204908758150592, −7.12889212776510277756148776096, −6.63380323174180755276159037395, −5.56983589302965558928171122032, −4.73357711413013048565129477522, −4.08049291419120175787386968306, −3.00953093497330524341368701081, −2.22291265811163532059722729414, −1.23860460411415196913081887870, 0,
1.23860460411415196913081887870, 2.22291265811163532059722729414, 3.00953093497330524341368701081, 4.08049291419120175787386968306, 4.73357711413013048565129477522, 5.56983589302965558928171122032, 6.63380323174180755276159037395, 7.12889212776510277756148776096, 7.82891765194818204908758150592
