| L(s) = 1 | + 2-s − 3-s + 4-s + 3.65·5-s − 6-s − 4.08·7-s + 8-s + 9-s + 3.65·10-s − 2.11·11-s − 12-s + 2.59·13-s − 4.08·14-s − 3.65·15-s + 16-s + 1.54·17-s + 18-s + 4.60·19-s + 3.65·20-s + 4.08·21-s − 2.11·22-s − 23-s − 24-s + 8.37·25-s + 2.59·26-s − 27-s − 4.08·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.63·5-s − 0.408·6-s − 1.54·7-s + 0.353·8-s + 0.333·9-s + 1.15·10-s − 0.636·11-s − 0.288·12-s + 0.719·13-s − 1.09·14-s − 0.944·15-s + 0.250·16-s + 0.375·17-s + 0.235·18-s + 1.05·19-s + 0.817·20-s + 0.890·21-s − 0.449·22-s − 0.208·23-s − 0.204·24-s + 1.67·25-s + 0.508·26-s − 0.192·27-s − 0.771·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.994852180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.994852180\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 - 3.65T + 5T^{2} \) |
| 7 | \( 1 + 4.08T + 7T^{2} \) |
| 11 | \( 1 + 2.11T + 11T^{2} \) |
| 13 | \( 1 - 2.59T + 13T^{2} \) |
| 17 | \( 1 - 1.54T + 17T^{2} \) |
| 19 | \( 1 - 4.60T + 19T^{2} \) |
| 31 | \( 1 - 0.201T + 31T^{2} \) |
| 37 | \( 1 - 4.67T + 37T^{2} \) |
| 41 | \( 1 - 1.23T + 41T^{2} \) |
| 43 | \( 1 + 3.50T + 43T^{2} \) |
| 47 | \( 1 + 2.84T + 47T^{2} \) |
| 53 | \( 1 - 7.19T + 53T^{2} \) |
| 59 | \( 1 - 9.32T + 59T^{2} \) |
| 61 | \( 1 - 4.11T + 61T^{2} \) |
| 67 | \( 1 - 1.70T + 67T^{2} \) |
| 71 | \( 1 - 4.12T + 71T^{2} \) |
| 73 | \( 1 - 4.28T + 73T^{2} \) |
| 79 | \( 1 + 5.51T + 79T^{2} \) |
| 83 | \( 1 + 0.151T + 83T^{2} \) |
| 89 | \( 1 - 8.36T + 89T^{2} \) |
| 97 | \( 1 + 1.09T + 97T^{2} \) |
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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
−8.525884678838935585765125896760, −7.34887392559514280117882047623, −6.66172696758325241560775003424, −6.02980401919331507715883655168, −5.65643835417496988725574523847, −4.98871716874039502037053751048, −3.74723102192508309599025427954, −2.99594556506652178256418450335, −2.15088068150512236490183329351, −0.933086844960607108684809475407,
0.933086844960607108684809475407, 2.15088068150512236490183329351, 2.99594556506652178256418450335, 3.74723102192508309599025427954, 4.98871716874039502037053751048, 5.65643835417496988725574523847, 6.02980401919331507715883655168, 6.66172696758325241560775003424, 7.34887392559514280117882047623, 8.525884678838935585765125896760
