L(s) = 1 | − 1.08·2-s + 3-s − 0.832·4-s + 5-s − 1.08·6-s + 2.50·7-s + 3.06·8-s + 9-s − 1.08·10-s + 3.09·11-s − 0.832·12-s + 5.05·13-s − 2.70·14-s + 15-s − 1.64·16-s − 4.41·17-s − 1.08·18-s − 8.37·19-s − 0.832·20-s + 2.50·21-s − 3.34·22-s + 3.06·24-s + 25-s − 5.46·26-s + 27-s − 2.08·28-s − 1.05·29-s + ⋯ |
L(s) = 1 | − 0.763·2-s + 0.577·3-s − 0.416·4-s + 0.447·5-s − 0.441·6-s + 0.945·7-s + 1.08·8-s + 0.333·9-s − 0.341·10-s + 0.934·11-s − 0.240·12-s + 1.40·13-s − 0.722·14-s + 0.258·15-s − 0.410·16-s − 1.07·17-s − 0.254·18-s − 1.92·19-s − 0.186·20-s + 0.545·21-s − 0.713·22-s + 0.624·24-s + 0.200·25-s − 1.07·26-s + 0.192·27-s − 0.393·28-s − 0.195·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.034013089\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.034013089\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + 1.08T + 2T^{2} \) |
| 7 | \( 1 - 2.50T + 7T^{2} \) |
| 11 | \( 1 - 3.09T + 11T^{2} \) |
| 13 | \( 1 - 5.05T + 13T^{2} \) |
| 17 | \( 1 + 4.41T + 17T^{2} \) |
| 19 | \( 1 + 8.37T + 19T^{2} \) |
| 29 | \( 1 + 1.05T + 29T^{2} \) |
| 31 | \( 1 + 5.62T + 31T^{2} \) |
| 37 | \( 1 - 6.45T + 37T^{2} \) |
| 41 | \( 1 - 3.14T + 41T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 - 8.62T + 47T^{2} \) |
| 53 | \( 1 + 9.34T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 12.0T + 61T^{2} \) |
| 67 | \( 1 + 7.01T + 67T^{2} \) |
| 71 | \( 1 + 6.26T + 71T^{2} \) |
| 73 | \( 1 - 15.4T + 73T^{2} \) |
| 79 | \( 1 - 6.74T + 79T^{2} \) |
| 83 | \( 1 - 14.2T + 83T^{2} \) |
| 89 | \( 1 - 5.93T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−8.008870291535560882996533287265, −7.45422361413727700876769058505, −6.44012424342966898567869043620, −6.00316888572295462837611112760, −4.80326420016457687895045729672, −4.22725858430761075514355318050, −3.74006652682804719575807359408, −2.25728776361499645495072143579, −1.73104136577077658105502321169, −0.824536481865939023888679072910,
0.824536481865939023888679072910, 1.73104136577077658105502321169, 2.25728776361499645495072143579, 3.74006652682804719575807359408, 4.22725858430761075514355318050, 4.80326420016457687895045729672, 6.00316888572295462837611112760, 6.44012424342966898567869043620, 7.45422361413727700876769058505, 8.008870291535560882996533287265