L(s) = 1 | − 1.73·2-s + 3-s + 0.999·4-s − 1.73·6-s + 3.46·7-s + 1.73·8-s + 9-s + 0.999·12-s + 1.73·13-s − 5.99·14-s − 5·16-s − 1.73·17-s − 1.73·18-s + 6.92·19-s + 3.46·21-s + 6·23-s + 1.73·24-s − 2.99·26-s + 27-s + 3.46·28-s − 1.73·29-s + 4·31-s + 5.19·32-s + 2.99·34-s + 0.999·36-s + 11·37-s − 11.9·38-s + ⋯ |
L(s) = 1 | − 1.22·2-s + 0.577·3-s + 0.499·4-s − 0.707·6-s + 1.30·7-s + 0.612·8-s + 0.333·9-s + 0.288·12-s + 0.480·13-s − 1.60·14-s − 1.25·16-s − 0.420·17-s − 0.408·18-s + 1.58·19-s + 0.755·21-s + 1.25·23-s + 0.353·24-s − 0.588·26-s + 0.192·27-s + 0.654·28-s − 0.321·29-s + 0.718·31-s + 0.918·32-s + 0.514·34-s + 0.166·36-s + 1.80·37-s − 1.94·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.893793142\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.893793142\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 13 | \( 1 - 1.73T + 13T^{2} \) |
| 17 | \( 1 + 1.73T + 17T^{2} \) |
| 19 | \( 1 - 6.92T + 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + 1.73T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 11T + 37T^{2} \) |
| 41 | \( 1 + 1.73T + 41T^{2} \) |
| 43 | \( 1 + 3.46T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 9T + 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 6.92T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 9T + 89T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
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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
−7.84605671860960842263485631020, −7.43292872227637892959162504220, −6.74544508007790591047393054058, −5.65654718699107272828865157544, −4.82935284143458366473936122396, −4.34908154766420695757309958265, −3.26783080166911657500046920693, −2.36809759390906627924853916185, −1.43276252738858661342632756536, −0.898254053855667962944737968862,
0.898254053855667962944737968862, 1.43276252738858661342632756536, 2.36809759390906627924853916185, 3.26783080166911657500046920693, 4.34908154766420695757309958265, 4.82935284143458366473936122396, 5.65654718699107272828865157544, 6.74544508007790591047393054058, 7.43292872227637892959162504220, 7.84605671860960842263485631020