L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 0.732·7-s + 8-s + 9-s + 3.73·11-s + 12-s + 2.73·13-s − 0.732·14-s + 16-s + 6.19·17-s + 18-s − 19-s − 0.732·21-s + 3.73·22-s − 5.92·23-s + 24-s + 2.73·26-s + 27-s − 0.732·28-s − 1.73·29-s − 2.46·31-s + 32-s + 3.73·33-s + 6.19·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.408·6-s − 0.276·7-s + 0.353·8-s + 0.333·9-s + 1.12·11-s + 0.288·12-s + 0.757·13-s − 0.195·14-s + 0.250·16-s + 1.50·17-s + 0.235·18-s − 0.229·19-s − 0.159·21-s + 0.795·22-s − 1.23·23-s + 0.204·24-s + 0.535·26-s + 0.192·27-s − 0.138·28-s − 0.321·29-s − 0.442·31-s + 0.176·32-s + 0.649·33-s + 1.06·34-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
\(L(1)\) |
\(\approx\) |
\(4.022577520\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.022577520\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.732T + 7T^{2} \) |
| 11 | \( 1 - 3.73T + 11T^{2} \) |
| 13 | \( 1 - 2.73T + 13T^{2} \) |
| 17 | \( 1 - 6.19T + 17T^{2} \) |
| 23 | \( 1 + 5.92T + 23T^{2} \) |
| 29 | \( 1 + 1.73T + 29T^{2} \) |
| 31 | \( 1 + 2.46T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 2.92T + 41T^{2} \) |
| 43 | \( 1 - 8.19T + 43T^{2} \) |
| 47 | \( 1 + 3.46T + 47T^{2} \) |
| 53 | \( 1 - 1.73T + 53T^{2} \) |
| 59 | \( 1 + 8.19T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 - 0.267T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 - 2.46T + 73T^{2} \) |
| 79 | \( 1 + 5.53T + 79T^{2} \) |
| 83 | \( 1 + 3.73T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 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
−8.726953230013388159642731996508, −7.965238822195098060537696902622, −7.26701332203973179580919135548, −6.30876323286851701940159826665, −5.86661213281215880643105074547, −4.76469398605454683043880058391, −3.69583606886267527919258304942, −3.52246917686238886789530786756, −2.21041592144861783304820456773, −1.20106690238198748164798732222,
1.20106690238198748164798732222, 2.21041592144861783304820456773, 3.52246917686238886789530786756, 3.69583606886267527919258304942, 4.76469398605454683043880058391, 5.86661213281215880643105074547, 6.30876323286851701940159826665, 7.26701332203973179580919135548, 7.965238822195098060537696902622, 8.726953230013388159642731996508