| L(s) = 1 | + 3-s − 1.06·5-s + 1.22·7-s + 9-s + 5.43·11-s + 6.43·13-s − 1.06·15-s − 1.04·17-s + 5.23·19-s + 1.22·21-s + 9.10·23-s − 3.86·25-s + 27-s − 1.65·29-s − 8.04·31-s + 5.43·33-s − 1.30·35-s + 10.1·37-s + 6.43·39-s − 5.53·41-s + 2.48·43-s − 1.06·45-s − 0.378·47-s − 5.48·49-s − 1.04·51-s − 3.42·53-s − 5.77·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.475·5-s + 0.464·7-s + 0.333·9-s + 1.63·11-s + 1.78·13-s − 0.274·15-s − 0.253·17-s + 1.20·19-s + 0.268·21-s + 1.89·23-s − 0.773·25-s + 0.192·27-s − 0.306·29-s − 1.44·31-s + 0.945·33-s − 0.221·35-s + 1.66·37-s + 1.03·39-s − 0.864·41-s + 0.379·43-s − 0.158·45-s − 0.0551·47-s − 0.783·49-s − 0.146·51-s − 0.469·53-s − 0.779·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.291481988\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.291481988\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 251 | \( 1 + T \) |
| good | 5 | \( 1 + 1.06T + 5T^{2} \) |
| 7 | \( 1 - 1.22T + 7T^{2} \) |
| 11 | \( 1 - 5.43T + 11T^{2} \) |
| 13 | \( 1 - 6.43T + 13T^{2} \) |
| 17 | \( 1 + 1.04T + 17T^{2} \) |
| 19 | \( 1 - 5.23T + 19T^{2} \) |
| 23 | \( 1 - 9.10T + 23T^{2} \) |
| 29 | \( 1 + 1.65T + 29T^{2} \) |
| 31 | \( 1 + 8.04T + 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 + 5.53T + 41T^{2} \) |
| 43 | \( 1 - 2.48T + 43T^{2} \) |
| 47 | \( 1 + 0.378T + 47T^{2} \) |
| 53 | \( 1 + 3.42T + 53T^{2} \) |
| 59 | \( 1 - 7.95T + 59T^{2} \) |
| 61 | \( 1 + 4.23T + 61T^{2} \) |
| 67 | \( 1 - 12.2T + 67T^{2} \) |
| 71 | \( 1 + 1.48T + 71T^{2} \) |
| 73 | \( 1 - 3.17T + 73T^{2} \) |
| 79 | \( 1 + 16.0T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 + 14.2T + 89T^{2} \) |
| 97 | \( 1 + 1.46T + 97T^{2} \) |
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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
−8.155297241794179078988449166977, −7.38746229397393414961561029023, −6.79200164138816885400882759171, −6.01836611394299857337244090455, −5.17376715593701497712623590240, −4.13406673106953161108824265357, −3.72440507940773236046497057128, −2.99888815811395130621484363675, −1.60369309773298843647408220941, −1.07492721843464835562214050625,
1.07492721843464835562214050625, 1.60369309773298843647408220941, 2.99888815811395130621484363675, 3.72440507940773236046497057128, 4.13406673106953161108824265357, 5.17376715593701497712623590240, 6.01836611394299857337244090455, 6.79200164138816885400882759171, 7.38746229397393414961561029023, 8.155297241794179078988449166977
