L(s) = 1 | − 2.33·2-s + 3.43·4-s − 1.49·5-s + 0.334·7-s − 3.33·8-s + 3.49·10-s + 3.28·11-s − 2.34·13-s − 0.779·14-s + 0.918·16-s − 4.10·17-s − 19-s − 5.14·20-s − 7.64·22-s − 4.14·23-s − 2.75·25-s + 5.45·26-s + 1.14·28-s − 8.24·29-s − 1.56·31-s + 4.53·32-s + 9.56·34-s − 0.501·35-s − 11.4·37-s + 2.33·38-s + 5.00·40-s − 1.31·41-s + ⋯ |
L(s) = 1 | − 1.64·2-s + 1.71·4-s − 0.669·5-s + 0.126·7-s − 1.18·8-s + 1.10·10-s + 0.989·11-s − 0.649·13-s − 0.208·14-s + 0.229·16-s − 0.994·17-s − 0.229·19-s − 1.14·20-s − 1.63·22-s − 0.865·23-s − 0.551·25-s + 1.07·26-s + 0.217·28-s − 1.53·29-s − 0.281·31-s + 0.802·32-s + 1.63·34-s − 0.0847·35-s − 1.88·37-s + 0.378·38-s + 0.791·40-s − 0.205·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2952165059\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2952165059\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| 47 | \( 1 + T \) |
good | 2 | \( 1 + 2.33T + 2T^{2} \) |
| 5 | \( 1 + 1.49T + 5T^{2} \) |
| 7 | \( 1 - 0.334T + 7T^{2} \) |
| 11 | \( 1 - 3.28T + 11T^{2} \) |
| 13 | \( 1 + 2.34T + 13T^{2} \) |
| 17 | \( 1 + 4.10T + 17T^{2} \) |
| 23 | \( 1 + 4.14T + 23T^{2} \) |
| 29 | \( 1 + 8.24T + 29T^{2} \) |
| 31 | \( 1 + 1.56T + 31T^{2} \) |
| 37 | \( 1 + 11.4T + 37T^{2} \) |
| 41 | \( 1 + 1.31T + 41T^{2} \) |
| 43 | \( 1 - 2.79T + 43T^{2} \) |
| 53 | \( 1 + 3.35T + 53T^{2} \) |
| 59 | \( 1 - 4.15T + 59T^{2} \) |
| 61 | \( 1 + 8.65T + 61T^{2} \) |
| 67 | \( 1 - 9.64T + 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 - 14.9T + 73T^{2} \) |
| 79 | \( 1 - 8.02T + 79T^{2} \) |
| 83 | \( 1 - 1.34T + 83T^{2} \) |
| 89 | \( 1 - 11.3T + 89T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
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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
−7.924531261111757967681275392608, −7.35615078390033600423261566598, −6.77361957586402243685041563229, −6.13447514293083008187028856528, −5.03594181006343604089575693613, −4.13788154210109329925152850667, −3.46095255322515700072505391760, −2.14171620542157960499970291318, −1.69522962794131038342985075123, −0.33825707244385100907871054325,
0.33825707244385100907871054325, 1.69522962794131038342985075123, 2.14171620542157960499970291318, 3.46095255322515700072505391760, 4.13788154210109329925152850667, 5.03594181006343604089575693613, 6.13447514293083008187028856528, 6.77361957586402243685041563229, 7.35615078390033600423261566598, 7.924531261111757967681275392608