| L(s) = 1 | + 2-s − 3-s + 4-s + 2.33·5-s − 6-s − 0.683·7-s + 8-s + 9-s + 2.33·10-s + 1.93·11-s − 12-s + 0.480·13-s − 0.683·14-s − 2.33·15-s + 16-s + 17-s + 18-s − 0.134·19-s + 2.33·20-s + 0.683·21-s + 1.93·22-s + 6.84·23-s − 24-s + 0.458·25-s + 0.480·26-s − 27-s − 0.683·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.04·5-s − 0.408·6-s − 0.258·7-s + 0.353·8-s + 0.333·9-s + 0.738·10-s + 0.584·11-s − 0.288·12-s + 0.133·13-s − 0.182·14-s − 0.603·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s − 0.0308·19-s + 0.522·20-s + 0.149·21-s + 0.412·22-s + 1.42·23-s − 0.204·24-s + 0.0916·25-s + 0.0942·26-s − 0.192·27-s − 0.129·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.567156949\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.567156949\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 5 | \( 1 - 2.33T + 5T^{2} \) |
| 7 | \( 1 + 0.683T + 7T^{2} \) |
| 11 | \( 1 - 1.93T + 11T^{2} \) |
| 13 | \( 1 - 0.480T + 13T^{2} \) |
| 19 | \( 1 + 0.134T + 19T^{2} \) |
| 23 | \( 1 - 6.84T + 23T^{2} \) |
| 29 | \( 1 - 7.79T + 29T^{2} \) |
| 31 | \( 1 - 2.80T + 31T^{2} \) |
| 37 | \( 1 + 0.709T + 37T^{2} \) |
| 41 | \( 1 + 2.73T + 41T^{2} \) |
| 43 | \( 1 + 6.44T + 43T^{2} \) |
| 47 | \( 1 + 8.90T + 47T^{2} \) |
| 53 | \( 1 - 13.4T + 53T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 - 8.92T + 67T^{2} \) |
| 71 | \( 1 - 5.82T + 71T^{2} \) |
| 73 | \( 1 + 2.58T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 + 5.79T + 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 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
−8.006710062182059505097019764760, −6.90487975100754759587925302328, −6.56246890955094395210977156277, −5.97560291942574682054109044253, −5.14285781528787081412371490506, −4.72744390129201548415370374059, −3.63909843008265364511246330057, −2.88021109956313785897858187817, −1.86285326087775444626233379169, −0.971771713180815008193905189964,
0.971771713180815008193905189964, 1.86285326087775444626233379169, 2.88021109956313785897858187817, 3.63909843008265364511246330057, 4.72744390129201548415370374059, 5.14285781528787081412371490506, 5.97560291942574682054109044253, 6.56246890955094395210977156277, 6.90487975100754759587925302328, 8.006710062182059505097019764760
