| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 3.16·7-s + 8-s + 9-s + 4.16·11-s − 12-s + 5.16·13-s + 3.16·14-s + 16-s + 0.837·17-s + 18-s − 19-s − 3.16·21-s + 4.16·22-s − 5.32·23-s − 24-s + 5.16·26-s − 27-s + 3.16·28-s − 3.83·29-s + 5.32·31-s + 32-s − 4.16·33-s + 0.837·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.408·6-s + 1.19·7-s + 0.353·8-s + 0.333·9-s + 1.25·11-s − 0.288·12-s + 1.43·13-s + 0.845·14-s + 0.250·16-s + 0.203·17-s + 0.235·18-s − 0.229·19-s − 0.690·21-s + 0.887·22-s − 1.11·23-s − 0.204·24-s + 1.01·26-s − 0.192·27-s + 0.597·28-s − 0.712·29-s + 0.956·31-s + 0.176·32-s − 0.724·33-s + 0.143·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.266897443\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.266897443\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 - 3.16T + 7T^{2} \) |
| 11 | \( 1 - 4.16T + 11T^{2} \) |
| 13 | \( 1 - 5.16T + 13T^{2} \) |
| 17 | \( 1 - 0.837T + 17T^{2} \) |
| 23 | \( 1 + 5.32T + 23T^{2} \) |
| 29 | \( 1 + 3.83T + 29T^{2} \) |
| 31 | \( 1 - 5.32T + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 + 6.32T + 41T^{2} \) |
| 43 | \( 1 + 9.16T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 - 1.83T + 53T^{2} \) |
| 59 | \( 1 + 7.48T + 59T^{2} \) |
| 61 | \( 1 + 4.16T + 61T^{2} \) |
| 67 | \( 1 + 6.48T + 67T^{2} \) |
| 71 | \( 1 - 1.16T + 71T^{2} \) |
| 73 | \( 1 + T + 73T^{2} \) |
| 79 | \( 1 + 5.32T + 79T^{2} \) |
| 83 | \( 1 - 6.48T + 83T^{2} \) |
| 89 | \( 1 + 7.32T + 89T^{2} \) |
| 97 | \( 1 - 0.513T + 97T^{2} \) |
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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
−8.591281669975302037044525835906, −8.032995874295812616320836664602, −7.10350260918887853504427632955, −6.16598670402954719517792420106, −5.89560895376279476522038578295, −4.74931527328241733480287833120, −4.20934265565401993689806227687, −3.41073646180013937204651051637, −1.89974976580143837146758565300, −1.17778454316745243393976271951,
1.17778454316745243393976271951, 1.89974976580143837146758565300, 3.41073646180013937204651051637, 4.20934265565401993689806227687, 4.74931527328241733480287833120, 5.89560895376279476522038578295, 6.16598670402954719517792420106, 7.10350260918887853504427632955, 8.032995874295812616320836664602, 8.591281669975302037044525835906
