| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 2.70·7-s − 8-s + 9-s + 5.62·11-s − 12-s + 4.84·13-s + 2.70·14-s + 16-s − 1.04·17-s − 18-s + 8.38·19-s + 2.70·21-s − 5.62·22-s + 5.29·23-s + 24-s − 4.84·26-s − 27-s − 2.70·28-s + 4.19·29-s − 5.58·31-s − 32-s − 5.62·33-s + 1.04·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s − 1.02·7-s − 0.353·8-s + 0.333·9-s + 1.69·11-s − 0.288·12-s + 1.34·13-s + 0.724·14-s + 0.250·16-s − 0.253·17-s − 0.235·18-s + 1.92·19-s + 0.591·21-s − 1.19·22-s + 1.10·23-s + 0.204·24-s − 0.949·26-s − 0.192·27-s − 0.511·28-s + 0.778·29-s − 1.00·31-s − 0.176·32-s − 0.978·33-s + 0.179·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.254156035\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.254156035\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2.70T + 7T^{2} \) |
| 11 | \( 1 - 5.62T + 11T^{2} \) |
| 13 | \( 1 - 4.84T + 13T^{2} \) |
| 17 | \( 1 + 1.04T + 17T^{2} \) |
| 19 | \( 1 - 8.38T + 19T^{2} \) |
| 23 | \( 1 - 5.29T + 23T^{2} \) |
| 29 | \( 1 - 4.19T + 29T^{2} \) |
| 31 | \( 1 + 5.58T + 31T^{2} \) |
| 37 | \( 1 - 2.99T + 37T^{2} \) |
| 41 | \( 1 + 1.57T + 41T^{2} \) |
| 43 | \( 1 + 3.29T + 43T^{2} \) |
| 47 | \( 1 - 3.07T + 47T^{2} \) |
| 53 | \( 1 + 6.17T + 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 - 0.270T + 61T^{2} \) |
| 67 | \( 1 + 6.30T + 67T^{2} \) |
| 71 | \( 1 + 4.53T + 71T^{2} \) |
| 73 | \( 1 - 8.16T + 73T^{2} \) |
| 79 | \( 1 - 12.4T + 79T^{2} \) |
| 83 | \( 1 - 0.134T + 83T^{2} \) |
| 89 | \( 1 - 11.7T + 89T^{2} \) |
| 97 | \( 1 - 2.64T + 97T^{2} \) |
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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
−8.781342872976324476469750764851, −7.69353272832598374117675486446, −6.91796974937162790875569275838, −6.39460978418062531121530163824, −5.86534414133804373394586337628, −4.77889291642387436387359308501, −3.60363212329171392478321715641, −3.17939728068629323199994087327, −1.52611911187121715581523877238, −0.827278709667328748558529711562,
0.827278709667328748558529711562, 1.52611911187121715581523877238, 3.17939728068629323199994087327, 3.60363212329171392478321715641, 4.77889291642387436387359308501, 5.86534414133804373394586337628, 6.39460978418062531121530163824, 6.91796974937162790875569275838, 7.69353272832598374117675486446, 8.781342872976324476469750764851
