| L(s) = 1 | − 0.343·2-s + 0.497·3-s − 1.88·4-s + 4.21·5-s − 0.170·6-s + 1.33·8-s − 2.75·9-s − 1.44·10-s − 0.252·11-s − 0.936·12-s + 6.96·13-s + 2.09·15-s + 3.30·16-s − 2.67·17-s + 0.945·18-s + 7.42·19-s − 7.93·20-s + 0.0866·22-s + 0.137·23-s + 0.663·24-s + 12.7·25-s − 2.39·26-s − 2.86·27-s − 2.53·29-s − 0.720·30-s − 2.20·31-s − 3.80·32-s + ⋯ |
| L(s) = 1 | − 0.242·2-s + 0.287·3-s − 0.940·4-s + 1.88·5-s − 0.0697·6-s + 0.471·8-s − 0.917·9-s − 0.457·10-s − 0.0760·11-s − 0.270·12-s + 1.93·13-s + 0.541·15-s + 0.826·16-s − 0.648·17-s + 0.222·18-s + 1.70·19-s − 1.77·20-s + 0.0184·22-s + 0.0287·23-s + 0.135·24-s + 2.55·25-s − 0.468·26-s − 0.550·27-s − 0.470·29-s − 0.131·30-s − 0.395·31-s − 0.672·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.523524341\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.523524341\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 + 0.343T + 2T^{2} \) |
| 3 | \( 1 - 0.497T + 3T^{2} \) |
| 5 | \( 1 - 4.21T + 5T^{2} \) |
| 11 | \( 1 + 0.252T + 11T^{2} \) |
| 13 | \( 1 - 6.96T + 13T^{2} \) |
| 17 | \( 1 + 2.67T + 17T^{2} \) |
| 19 | \( 1 - 7.42T + 19T^{2} \) |
| 23 | \( 1 - 0.137T + 23T^{2} \) |
| 29 | \( 1 + 2.53T + 29T^{2} \) |
| 31 | \( 1 + 2.20T + 31T^{2} \) |
| 37 | \( 1 - 7.40T + 37T^{2} \) |
| 41 | \( 1 - 4.80T + 41T^{2} \) |
| 43 | \( 1 - 2.71T + 43T^{2} \) |
| 47 | \( 1 + 5.23T + 47T^{2} \) |
| 53 | \( 1 - 7.52T + 53T^{2} \) |
| 59 | \( 1 + 5.54T + 59T^{2} \) |
| 61 | \( 1 + 0.883T + 61T^{2} \) |
| 67 | \( 1 + 15.1T + 67T^{2} \) |
| 71 | \( 1 - 6.41T + 71T^{2} \) |
| 73 | \( 1 + 5.39T + 73T^{2} \) |
| 79 | \( 1 + 9.73T + 79T^{2} \) |
| 83 | \( 1 - 4.76T + 83T^{2} \) |
| 89 | \( 1 - 7.64T + 89T^{2} \) |
| 97 | \( 1 - 0.551T + 97T^{2} \) |
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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
−8.308269676361012975287267543412, −7.48565795286677254880535526537, −6.37899622137603402074359665805, −5.75819362580643923757641198941, −5.50012536826254453858187049826, −4.50713254080005164021461719865, −3.48278176520898595579229662939, −2.77228150835345378511955237527, −1.68264801970544358720336098740, −0.926504864016263825441239624364,
0.926504864016263825441239624364, 1.68264801970544358720336098740, 2.77228150835345378511955237527, 3.48278176520898595579229662939, 4.50713254080005164021461719865, 5.50012536826254453858187049826, 5.75819362580643923757641198941, 6.37899622137603402074359665805, 7.48565795286677254880535526537, 8.308269676361012975287267543412
