| L(s) = 1 | + 3-s + 4.14·5-s − 3.34·7-s + 9-s − 6.37·11-s − 1.56·13-s + 4.14·15-s + 3.05·17-s − 2.45·19-s − 3.34·21-s + 4.22·23-s + 12.1·25-s + 27-s + 8.04·29-s + 0.582·31-s − 6.37·33-s − 13.8·35-s + 7.18·37-s − 1.56·39-s − 6.06·41-s + 4.77·43-s + 4.14·45-s − 3.66·47-s + 4.19·49-s + 3.05·51-s + 11.6·53-s − 26.4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.85·5-s − 1.26·7-s + 0.333·9-s − 1.92·11-s − 0.434·13-s + 1.07·15-s + 0.741·17-s − 0.563·19-s − 0.730·21-s + 0.881·23-s + 2.43·25-s + 0.192·27-s + 1.49·29-s + 0.104·31-s − 1.11·33-s − 2.34·35-s + 1.18·37-s − 0.251·39-s − 0.947·41-s + 0.727·43-s + 0.618·45-s − 0.534·47-s + 0.599·49-s + 0.428·51-s + 1.60·53-s − 3.56·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.815281596\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.815281596\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 503 | \( 1 - T \) |
| good | 5 | \( 1 - 4.14T + 5T^{2} \) |
| 7 | \( 1 + 3.34T + 7T^{2} \) |
| 11 | \( 1 + 6.37T + 11T^{2} \) |
| 13 | \( 1 + 1.56T + 13T^{2} \) |
| 17 | \( 1 - 3.05T + 17T^{2} \) |
| 19 | \( 1 + 2.45T + 19T^{2} \) |
| 23 | \( 1 - 4.22T + 23T^{2} \) |
| 29 | \( 1 - 8.04T + 29T^{2} \) |
| 31 | \( 1 - 0.582T + 31T^{2} \) |
| 37 | \( 1 - 7.18T + 37T^{2} \) |
| 41 | \( 1 + 6.06T + 41T^{2} \) |
| 43 | \( 1 - 4.77T + 43T^{2} \) |
| 47 | \( 1 + 3.66T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 - 11.9T + 59T^{2} \) |
| 61 | \( 1 - 0.0177T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 + 3.10T + 71T^{2} \) |
| 73 | \( 1 - 7.45T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 + 4.48T + 83T^{2} \) |
| 89 | \( 1 + 11.5T + 89T^{2} \) |
| 97 | \( 1 + 3.01T + 97T^{2} \) |
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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
−8.251862901384187046635388135749, −7.20740945724690957731427525089, −6.68681850324308973077559806686, −5.86304149118837258583417460135, −5.37066088031354756549418774039, −4.61559672686664852127419698437, −3.22101397655555028632987764923, −2.67161377242431140715369603470, −2.22307107384217614421456144438, −0.836552402267532679128530613475,
0.836552402267532679128530613475, 2.22307107384217614421456144438, 2.67161377242431140715369603470, 3.22101397655555028632987764923, 4.61559672686664852127419698437, 5.37066088031354756549418774039, 5.86304149118837258583417460135, 6.68681850324308973077559806686, 7.20740945724690957731427525089, 8.251862901384187046635388135749
