| L(s) = 1 | − 2-s + 1.61·3-s + 4-s − 0.618·5-s − 1.61·6-s − 7-s − 8-s − 0.381·9-s + 0.618·10-s + 1.85·11-s + 1.61·12-s − 2.38·13-s + 14-s − 1.00·15-s + 16-s + 5.47·17-s + 0.381·18-s − 7.47·19-s − 0.618·20-s − 1.61·21-s − 1.85·22-s − 1.61·24-s − 4.61·25-s + 2.38·26-s − 5.47·27-s − 28-s + 7.70·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.934·3-s + 0.5·4-s − 0.276·5-s − 0.660·6-s − 0.377·7-s − 0.353·8-s − 0.127·9-s + 0.195·10-s + 0.559·11-s + 0.467·12-s − 0.660·13-s + 0.267·14-s − 0.258·15-s + 0.250·16-s + 1.32·17-s + 0.0900·18-s − 1.71·19-s − 0.138·20-s − 0.353·21-s − 0.395·22-s − 0.330·24-s − 0.923·25-s + 0.467·26-s − 1.05·27-s − 0.188·28-s + 1.43·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.436830557\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.436830557\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 1.61T + 3T^{2} \) |
| 5 | \( 1 + 0.618T + 5T^{2} \) |
| 11 | \( 1 - 1.85T + 11T^{2} \) |
| 13 | \( 1 + 2.38T + 13T^{2} \) |
| 17 | \( 1 - 5.47T + 17T^{2} \) |
| 19 | \( 1 + 7.47T + 19T^{2} \) |
| 29 | \( 1 - 7.70T + 29T^{2} \) |
| 31 | \( 1 + 7.94T + 31T^{2} \) |
| 37 | \( 1 + 3T + 37T^{2} \) |
| 41 | \( 1 - 6.85T + 41T^{2} \) |
| 43 | \( 1 + 3.47T + 43T^{2} \) |
| 47 | \( 1 - 4.09T + 47T^{2} \) |
| 53 | \( 1 - 9.70T + 53T^{2} \) |
| 59 | \( 1 + 6.76T + 59T^{2} \) |
| 61 | \( 1 - 5.70T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 - 9.09T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 + 1.47T + 79T^{2} \) |
| 83 | \( 1 + 1.38T + 83T^{2} \) |
| 89 | \( 1 - 9.94T + 89T^{2} \) |
| 97 | \( 1 - 13.4T + 97T^{2} \) |
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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
−7.934330334338469851048764043705, −7.49762237533026356022529849921, −6.66214204181980656107864205310, −6.00863437123453080466942046171, −5.13773703434831341625261139656, −3.95415732435534057524469056416, −3.51823507569711270613803515531, −2.53053079110439674204915437053, −1.95869495461814122459887104696, −0.61594560660471592589664532325,
0.61594560660471592589664532325, 1.95869495461814122459887104696, 2.53053079110439674204915437053, 3.51823507569711270613803515531, 3.95415732435534057524469056416, 5.13773703434831341625261139656, 6.00863437123453080466942046171, 6.66214204181980656107864205310, 7.49762237533026356022529849921, 7.934330334338469851048764043705
