| L(s) = 1 | − 0.536·2-s − 3-s − 1.71·4-s + 0.536·6-s − 2.57·7-s + 1.99·8-s + 9-s − 5.14·11-s + 1.71·12-s + 1.47·13-s + 1.38·14-s + 2.35·16-s − 0.687·17-s − 0.536·18-s − 8.09·19-s + 2.57·21-s + 2.75·22-s − 0.372·23-s − 1.99·24-s − 0.791·26-s − 27-s + 4.40·28-s + 0.0356·29-s − 4.48·31-s − 5.24·32-s + 5.14·33-s + 0.369·34-s + ⋯ |
| L(s) = 1 | − 0.379·2-s − 0.577·3-s − 0.856·4-s + 0.219·6-s − 0.972·7-s + 0.704·8-s + 0.333·9-s − 1.55·11-s + 0.494·12-s + 0.409·13-s + 0.368·14-s + 0.588·16-s − 0.166·17-s − 0.126·18-s − 1.85·19-s + 0.561·21-s + 0.588·22-s − 0.0777·23-s − 0.406·24-s − 0.155·26-s − 0.192·27-s + 0.832·28-s + 0.00662·29-s − 0.806·31-s − 0.927·32-s + 0.895·33-s + 0.0632·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3419758708\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3419758708\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 0.536T + 2T^{2} \) |
| 7 | \( 1 + 2.57T + 7T^{2} \) |
| 11 | \( 1 + 5.14T + 11T^{2} \) |
| 13 | \( 1 - 1.47T + 13T^{2} \) |
| 17 | \( 1 + 0.687T + 17T^{2} \) |
| 19 | \( 1 + 8.09T + 19T^{2} \) |
| 23 | \( 1 + 0.372T + 23T^{2} \) |
| 29 | \( 1 - 0.0356T + 29T^{2} \) |
| 31 | \( 1 + 4.48T + 31T^{2} \) |
| 37 | \( 1 + 1.90T + 37T^{2} \) |
| 41 | \( 1 + 5.16T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 + 8.52T + 47T^{2} \) |
| 53 | \( 1 - 9.12T + 53T^{2} \) |
| 59 | \( 1 + 0.176T + 59T^{2} \) |
| 61 | \( 1 + 6.41T + 61T^{2} \) |
| 67 | \( 1 + 0.0834T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 - 4.95T + 79T^{2} \) |
| 83 | \( 1 + 9.36T + 83T^{2} \) |
| 89 | \( 1 + 0.0123T + 89T^{2} \) |
| 97 | \( 1 - 7.62T + 97T^{2} \) |
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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
−9.245996322123426313623360551008, −8.496362670103204957873514329330, −7.80504243036636994411857753160, −6.83357419487943998865911384487, −5.99383232878288142116114930915, −5.21551793763262009442878124800, −4.36469720209571026985882235881, −3.44125351403919307805852314845, −2.13051149110699871948062925657, −0.41277562080690542901938272003,
0.41277562080690542901938272003, 2.13051149110699871948062925657, 3.44125351403919307805852314845, 4.36469720209571026985882235881, 5.21551793763262009442878124800, 5.99383232878288142116114930915, 6.83357419487943998865911384487, 7.80504243036636994411857753160, 8.496362670103204957873514329330, 9.245996322123426313623360551008
