| L(s) = 1 | + 2-s + 2.31·3-s + 4-s + 4.15·5-s + 2.31·6-s − 1.57·7-s + 8-s + 2.35·9-s + 4.15·10-s + 2.35·11-s + 2.31·12-s − 3.92·13-s − 1.57·14-s + 9.61·15-s + 16-s + 3.89·17-s + 2.35·18-s − 4.83·19-s + 4.15·20-s − 3.64·21-s + 2.35·22-s + 23-s + 2.31·24-s + 12.2·25-s − 3.92·26-s − 1.49·27-s − 1.57·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.33·3-s + 0.5·4-s + 1.85·5-s + 0.944·6-s − 0.595·7-s + 0.353·8-s + 0.784·9-s + 1.31·10-s + 0.709·11-s + 0.667·12-s − 1.08·13-s − 0.420·14-s + 2.48·15-s + 0.250·16-s + 0.944·17-s + 0.554·18-s − 1.10·19-s + 0.929·20-s − 0.795·21-s + 0.501·22-s + 0.208·23-s + 0.472·24-s + 2.45·25-s − 0.769·26-s − 0.287·27-s − 0.297·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.137020824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.137020824\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 - 2.31T + 3T^{2} \) |
| 5 | \( 1 - 4.15T + 5T^{2} \) |
| 7 | \( 1 + 1.57T + 7T^{2} \) |
| 11 | \( 1 - 2.35T + 11T^{2} \) |
| 13 | \( 1 + 3.92T + 13T^{2} \) |
| 17 | \( 1 - 3.89T + 17T^{2} \) |
| 19 | \( 1 + 4.83T + 19T^{2} \) |
| 29 | \( 1 + 4.11T + 29T^{2} \) |
| 31 | \( 1 - 4.26T + 31T^{2} \) |
| 37 | \( 1 + 0.669T + 37T^{2} \) |
| 41 | \( 1 - 7.10T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 + 6.63T + 47T^{2} \) |
| 53 | \( 1 - 7.32T + 53T^{2} \) |
| 59 | \( 1 - 7.42T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 - 0.236T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 - 3.35T + 73T^{2} \) |
| 79 | \( 1 - 7.78T + 79T^{2} \) |
| 83 | \( 1 - 6.70T + 83T^{2} \) |
| 89 | \( 1 + 14.9T + 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
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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
−8.106886742991454692317187495380, −7.21766879495891898665415952255, −6.59719443271508040654791165274, −5.90614764240540186387191194310, −5.31137130185684107694871757901, −4.32171688847171584237906176722, −3.50158757003962077660466053394, −2.53458476996562366538992541935, −2.36823533392064788288052211935, −1.31568688520929069779232879783,
1.31568688520929069779232879783, 2.36823533392064788288052211935, 2.53458476996562366538992541935, 3.50158757003962077660466053394, 4.32171688847171584237906176722, 5.31137130185684107694871757901, 5.90614764240540186387191194310, 6.59719443271508040654791165274, 7.21766879495891898665415952255, 8.106886742991454692317187495380
