L(s) = 1 | − 2.62·2-s − 0.903·3-s + 4.90·4-s + 0.337·5-s + 2.37·6-s − 7.63·8-s − 2.18·9-s − 0.888·10-s − 4.43·12-s − 6.09·13-s − 0.305·15-s + 10.2·16-s − 3.93·17-s + 5.74·18-s + 8.10·19-s + 1.65·20-s + 1.82·23-s + 6.90·24-s − 4.88·25-s + 16.0·26-s + 4.68·27-s + 6.61·29-s + 0.802·30-s − 6.18·31-s − 11.6·32-s + 10.3·34-s − 10.7·36-s + ⋯ |
L(s) = 1 | − 1.85·2-s − 0.521·3-s + 2.45·4-s + 0.151·5-s + 0.969·6-s − 2.70·8-s − 0.728·9-s − 0.280·10-s − 1.27·12-s − 1.68·13-s − 0.0788·15-s + 2.56·16-s − 0.955·17-s + 1.35·18-s + 1.86·19-s + 0.370·20-s + 0.381·23-s + 1.40·24-s − 0.977·25-s + 3.13·26-s + 0.901·27-s + 1.22·29-s + 0.146·30-s − 1.11·31-s − 2.06·32-s + 1.77·34-s − 1.78·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2880063281\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2880063281\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 2.62T + 2T^{2} \) |
| 3 | \( 1 + 0.903T + 3T^{2} \) |
| 5 | \( 1 - 0.337T + 5T^{2} \) |
| 13 | \( 1 + 6.09T + 13T^{2} \) |
| 17 | \( 1 + 3.93T + 17T^{2} \) |
| 19 | \( 1 - 8.10T + 19T^{2} \) |
| 23 | \( 1 - 1.82T + 23T^{2} \) |
| 29 | \( 1 - 6.61T + 29T^{2} \) |
| 31 | \( 1 + 6.18T + 31T^{2} \) |
| 37 | \( 1 + 0.706T + 37T^{2} \) |
| 41 | \( 1 - 1.45T + 41T^{2} \) |
| 43 | \( 1 + 2.45T + 43T^{2} \) |
| 47 | \( 1 + 4.70T + 47T^{2} \) |
| 53 | \( 1 - 6.15T + 53T^{2} \) |
| 59 | \( 1 + 6.77T + 59T^{2} \) |
| 61 | \( 1 - 4.28T + 61T^{2} \) |
| 67 | \( 1 + 8.68T + 67T^{2} \) |
| 71 | \( 1 + 4.68T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 - 0.337T + 89T^{2} \) |
| 97 | \( 1 - 6.74T + 97T^{2} \) |
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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
−8.171769470324916921382503655986, −7.31708509533764789579008470855, −7.13491042516225072804896828495, −6.15557371848905369490425650617, −5.49588531396184111829278028259, −4.67935897247714656676628890031, −3.10962547239122395828090033019, −2.52826392242055658596974520906, −1.54562647554109821987402238992, −0.38253691857464134751482705968,
0.38253691857464134751482705968, 1.54562647554109821987402238992, 2.52826392242055658596974520906, 3.10962547239122395828090033019, 4.67935897247714656676628890031, 5.49588531396184111829278028259, 6.15557371848905369490425650617, 7.13491042516225072804896828495, 7.31708509533764789579008470855, 8.171769470324916921382503655986