| L(s) = 1 | − 0.774·3-s − 2.66·5-s + 0.769·7-s − 2.40·9-s + 11-s − 5.64·13-s + 2.06·15-s + 7.62·17-s − 4.37·19-s − 0.595·21-s − 5.32·23-s + 2.10·25-s + 4.18·27-s − 5.41·29-s + 7.12·31-s − 0.774·33-s − 2.04·35-s − 5.67·37-s + 4.37·39-s + 1.11·41-s − 1.92·43-s + 6.39·45-s − 10.8·47-s − 6.40·49-s − 5.90·51-s − 0.282·53-s − 2.66·55-s + ⋯ |
| L(s) = 1 | − 0.447·3-s − 1.19·5-s + 0.290·7-s − 0.800·9-s + 0.301·11-s − 1.56·13-s + 0.532·15-s + 1.84·17-s − 1.00·19-s − 0.129·21-s − 1.11·23-s + 0.420·25-s + 0.804·27-s − 1.00·29-s + 1.27·31-s − 0.134·33-s − 0.346·35-s − 0.932·37-s + 0.700·39-s + 0.174·41-s − 0.294·43-s + 0.953·45-s − 1.58·47-s − 0.915·49-s − 0.826·51-s − 0.0388·53-s − 0.359·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5252673296\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5252673296\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 137 | \( 1 - T \) |
| good | 3 | \( 1 + 0.774T + 3T^{2} \) |
| 5 | \( 1 + 2.66T + 5T^{2} \) |
| 7 | \( 1 - 0.769T + 7T^{2} \) |
| 13 | \( 1 + 5.64T + 13T^{2} \) |
| 17 | \( 1 - 7.62T + 17T^{2} \) |
| 19 | \( 1 + 4.37T + 19T^{2} \) |
| 23 | \( 1 + 5.32T + 23T^{2} \) |
| 29 | \( 1 + 5.41T + 29T^{2} \) |
| 31 | \( 1 - 7.12T + 31T^{2} \) |
| 37 | \( 1 + 5.67T + 37T^{2} \) |
| 41 | \( 1 - 1.11T + 41T^{2} \) |
| 43 | \( 1 + 1.92T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 + 0.282T + 53T^{2} \) |
| 59 | \( 1 + 12.6T + 59T^{2} \) |
| 61 | \( 1 - 9.53T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 + 8.75T + 73T^{2} \) |
| 79 | \( 1 - 5.68T + 79T^{2} \) |
| 83 | \( 1 + 6.98T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 - 8.05T + 97T^{2} \) |
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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
−8.017359125854643383208846118776, −7.54449005015802758034250498972, −6.68768493827312690764567983885, −5.90360525830494368430620007730, −5.13552639447900775235599158194, −4.52741230326813558675224145248, −3.64566388016090087914017888387, −2.93737847285284838201095201580, −1.80517487985373237565887020820, −0.37612850196355209122760192148,
0.37612850196355209122760192148, 1.80517487985373237565887020820, 2.93737847285284838201095201580, 3.64566388016090087914017888387, 4.52741230326813558675224145248, 5.13552639447900775235599158194, 5.90360525830494368430620007730, 6.68768493827312690764567983885, 7.54449005015802758034250498972, 8.017359125854643383208846118776
