| L(s) = 1 | + 1.26·3-s + 3.64·5-s − 1.51·7-s − 1.40·9-s + 1.84·11-s + 3.46·13-s + 4.60·15-s + 1.23·17-s − 4.85·19-s − 1.91·21-s + 8.04·23-s + 8.25·25-s − 5.56·27-s + 29-s − 5.45·31-s + 2.32·33-s − 5.51·35-s + 11.9·37-s + 4.37·39-s + 3.01·41-s + 1.84·43-s − 5.10·45-s + 11.3·47-s − 4.70·49-s + 1.56·51-s + 0.535·53-s + 6.70·55-s + ⋯ |
| L(s) = 1 | + 0.729·3-s + 1.62·5-s − 0.572·7-s − 0.467·9-s + 0.555·11-s + 0.960·13-s + 1.18·15-s + 0.300·17-s − 1.11·19-s − 0.417·21-s + 1.67·23-s + 1.65·25-s − 1.07·27-s + 0.185·29-s − 0.979·31-s + 0.405·33-s − 0.932·35-s + 1.96·37-s + 0.701·39-s + 0.470·41-s + 0.280·43-s − 0.760·45-s + 1.65·47-s − 0.672·49-s + 0.219·51-s + 0.0736·53-s + 0.903·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.957445346\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.957445346\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 1.26T + 3T^{2} \) |
| 5 | \( 1 - 3.64T + 5T^{2} \) |
| 7 | \( 1 + 1.51T + 7T^{2} \) |
| 11 | \( 1 - 1.84T + 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 - 1.23T + 17T^{2} \) |
| 19 | \( 1 + 4.85T + 19T^{2} \) |
| 23 | \( 1 - 8.04T + 23T^{2} \) |
| 31 | \( 1 + 5.45T + 31T^{2} \) |
| 37 | \( 1 - 11.9T + 37T^{2} \) |
| 41 | \( 1 - 3.01T + 41T^{2} \) |
| 43 | \( 1 - 1.84T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 - 0.535T + 53T^{2} \) |
| 59 | \( 1 + 5.01T + 59T^{2} \) |
| 61 | \( 1 + 9.32T + 61T^{2} \) |
| 67 | \( 1 + 2.67T + 67T^{2} \) |
| 71 | \( 1 - 8.03T + 71T^{2} \) |
| 73 | \( 1 - 9.48T + 73T^{2} \) |
| 79 | \( 1 + 16.4T + 79T^{2} \) |
| 83 | \( 1 - 2.71T + 83T^{2} \) |
| 89 | \( 1 + 3.19T + 89T^{2} \) |
| 97 | \( 1 - 5.68T + 97T^{2} \) |
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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
−9.093032273049499112807495118607, −8.880893763195023017802551881795, −7.76600432301642592673245039751, −6.62840814338440210249146929812, −6.10492364405449883525275883138, −5.43296070481913198369081303455, −4.15103554493679446308620313113, −3.07988272261199512728357272830, −2.37621837718280977700492385315, −1.24728636001167613529215300865,
1.24728636001167613529215300865, 2.37621837718280977700492385315, 3.07988272261199512728357272830, 4.15103554493679446308620313113, 5.43296070481913198369081303455, 6.10492364405449883525275883138, 6.62840814338440210249146929812, 7.76600432301642592673245039751, 8.880893763195023017802551881795, 9.093032273049499112807495118607
