| 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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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
−8.871601244036748557495100453618, −8.177461918242554390582217876560, −7.15866437729895619340684604655, −6.20486576122277783089917966107, −5.63062637187271031613866358892, −4.82168151239079328090634807749, −3.76106155226962667586980804739, −2.78415746848627045879440999997, −1.48382369483790907588795413592, 0,
1.48382369483790907588795413592, 2.78415746848627045879440999997, 3.76106155226962667586980804739, 4.82168151239079328090634807749, 5.63062637187271031613866358892, 6.20486576122277783089917966107, 7.15866437729895619340684604655, 8.177461918242554390582217876560, 8.871601244036748557495100453618
