| L(s) = 1 | − 2-s + 4-s − 8-s − 6·11-s − 13-s + 16-s + 7·17-s − 7·19-s + 6·22-s − 8·23-s + 26-s + 2·29-s + 8·31-s − 32-s − 7·34-s + 8·37-s + 7·38-s − 3·41-s + 7·43-s − 6·44-s + 8·46-s − 10·47-s − 52-s − 12·53-s − 2·58-s + 4·59-s + 12·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.80·11-s − 0.277·13-s + 1/4·16-s + 1.69·17-s − 1.60·19-s + 1.27·22-s − 1.66·23-s + 0.196·26-s + 0.371·29-s + 1.43·31-s − 0.176·32-s − 1.20·34-s + 1.31·37-s + 1.13·38-s − 0.468·41-s + 1.06·43-s − 0.904·44-s + 1.17·46-s − 1.45·47-s − 0.138·52-s − 1.64·53-s − 0.262·58-s + 0.520·59-s + 1.53·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−12.81545127513858, −12.48749963237348, −12.03151468688364, −11.60546486630170, −10.83061191041063, −10.69360091531861, −10.11098848539915, −9.731690737291384, −9.593962981261139, −8.544641510501919, −8.172058351202177, −7.988736293635990, −7.661979853361721, −6.906555247713215, −6.345499944128662, −5.965020296532456, −5.462062145824207, −4.858325495458860, −4.400033428447502, −3.688655842733930, −3.061070413845081, −2.485579802038974, −2.195644320167199, −1.385506391237972, −0.6180286696150423, 0,
0.6180286696150423, 1.385506391237972, 2.195644320167199, 2.485579802038974, 3.061070413845081, 3.688655842733930, 4.400033428447502, 4.858325495458860, 5.462062145824207, 5.965020296532456, 6.345499944128662, 6.906555247713215, 7.661979853361721, 7.988736293635990, 8.172058351202177, 8.544641510501919, 9.593962981261139, 9.731690737291384, 10.11098848539915, 10.69360091531861, 10.83061191041063, 11.60546486630170, 12.03151468688364, 12.48749963237348, 12.81545127513858
