| L(s) = 1 | − 2·2-s + 3.16·3-s + 4·4-s − 6.32·6-s − 3.52·7-s − 8·8-s − 16.9·9-s − 16.9·11-s + 12.6·12-s − 46.0·13-s + 7.04·14-s + 16·16-s − 16.2·17-s + 33.9·18-s − 31.1·19-s − 11.1·21-s + 33.9·22-s + 23·23-s − 25.3·24-s + 92.0·26-s − 139.·27-s − 14.0·28-s − 11.4·29-s − 82.7·31-s − 32·32-s − 53.6·33-s + 32.5·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.608·3-s + 0.5·4-s − 0.430·6-s − 0.190·7-s − 0.353·8-s − 0.629·9-s − 0.465·11-s + 0.304·12-s − 0.981·13-s + 0.134·14-s + 0.250·16-s − 0.231·17-s + 0.445·18-s − 0.375·19-s − 0.115·21-s + 0.328·22-s + 0.208·23-s − 0.215·24-s + 0.694·26-s − 0.991·27-s − 0.0951·28-s − 0.0730·29-s − 0.479·31-s − 0.176·32-s − 0.283·33-s + 0.164·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.199198396\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.199198396\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 5 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 - 3.16T + 27T^{2} \) |
| 7 | \( 1 + 3.52T + 343T^{2} \) |
| 11 | \( 1 + 16.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 46.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 16.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 31.1T + 6.85e3T^{2} \) |
| 29 | \( 1 + 11.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 82.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 296.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 407.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 549.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 503.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 605.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 522.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 242.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.01e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 334.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 181.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 461.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 817.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 774.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.40e3T + 9.12e5T^{2} \) |
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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
−9.409383316082517112100589200625, −8.628370678697078218153770832048, −7.88441557286766926519377367546, −7.23230895265299639827708349393, −6.17695980928865396913159555886, −5.26815124815602865955580788353, −4.02118317602473161574000774678, −2.78385303650012838736972539211, −2.22906457414017634777173715430, −0.57969260093478085053477405992,
0.57969260093478085053477405992, 2.22906457414017634777173715430, 2.78385303650012838736972539211, 4.02118317602473161574000774678, 5.26815124815602865955580788353, 6.17695980928865396913159555886, 7.23230895265299639827708349393, 7.88441557286766926519377367546, 8.628370678697078218153770832048, 9.409383316082517112100589200625
