L(s) = 1 | + 2.61·2-s − 3-s + 4.85·4-s + 1.61·5-s − 2.61·6-s + 7-s + 7.47·8-s + 9-s + 4.23·10-s − 4.85·12-s + 5.47·13-s + 2.61·14-s − 1.61·15-s + 9.85·16-s − 3·17-s + 2.61·18-s + 4·19-s + 7.85·20-s − 21-s − 7.23·23-s − 7.47·24-s − 2.38·25-s + 14.3·26-s − 27-s + 4.85·28-s + 3·29-s − 4.23·30-s + ⋯ |
L(s) = 1 | + 1.85·2-s − 0.577·3-s + 2.42·4-s + 0.723·5-s − 1.06·6-s + 0.377·7-s + 2.64·8-s + 0.333·9-s + 1.33·10-s − 1.40·12-s + 1.51·13-s + 0.699·14-s − 0.417·15-s + 2.46·16-s − 0.727·17-s + 0.617·18-s + 0.917·19-s + 1.75·20-s − 0.218·21-s − 1.50·23-s − 1.52·24-s − 0.476·25-s + 2.80·26-s − 0.192·27-s + 0.917·28-s + 0.557·29-s − 0.773·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.134157940\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.134157940\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 2.61T + 2T^{2} \) |
| 5 | \( 1 - 1.61T + 5T^{2} \) |
| 13 | \( 1 - 5.47T + 13T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 7.23T + 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 - 0.236T + 31T^{2} \) |
| 37 | \( 1 + 8.47T + 37T^{2} \) |
| 41 | \( 1 + 3.09T + 41T^{2} \) |
| 43 | \( 1 + 9T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 - 4.09T + 53T^{2} \) |
| 59 | \( 1 - 12.6T + 59T^{2} \) |
| 61 | \( 1 + 6.70T + 61T^{2} \) |
| 67 | \( 1 + 3.70T + 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 + 3.85T + 73T^{2} \) |
| 79 | \( 1 - 6.85T + 79T^{2} \) |
| 83 | \( 1 - 2.94T + 83T^{2} \) |
| 89 | \( 1 - 2.67T + 89T^{2} \) |
| 97 | \( 1 + 16.4T + 97T^{2} \) |
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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
−8.818206925268078264570657136605, −7.86859943884826583841346149544, −6.83158605063374227427690695613, −6.31100782834095102140711766073, −5.62417421125164964876499184476, −5.13987419869278647024153406567, −4.11977471639089980700279979599, −3.54489809330627932369929671022, −2.28156565720515830357041089702, −1.46840296827270081197226849817,
1.46840296827270081197226849817, 2.28156565720515830357041089702, 3.54489809330627932369929671022, 4.11977471639089980700279979599, 5.13987419869278647024153406567, 5.62417421125164964876499184476, 6.31100782834095102140711766073, 6.83158605063374227427690695613, 7.86859943884826583841346149544, 8.818206925268078264570657136605