| L(s) = 1 | + 2-s + 4-s + 8-s − 2·11-s − 13-s + 16-s − 2·17-s + 5·19-s − 2·22-s − 6·23-s − 26-s + 5·29-s + 8·31-s + 32-s − 2·34-s − 7·37-s + 5·38-s + 3·41-s + 8·43-s − 2·44-s − 6·46-s + 4·47-s − 7·49-s − 52-s − 8·53-s + 5·58-s + 9·59-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 0.603·11-s − 0.277·13-s + 1/4·16-s − 0.485·17-s + 1.14·19-s − 0.426·22-s − 1.25·23-s − 0.196·26-s + 0.928·29-s + 1.43·31-s + 0.176·32-s − 0.342·34-s − 1.15·37-s + 0.811·38-s + 0.468·41-s + 1.21·43-s − 0.301·44-s − 0.884·46-s + 0.583·47-s − 49-s − 0.138·52-s − 1.09·53-s + 0.656·58-s + 1.17·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.09410436402507, −13.81199799204114, −13.55306398572525, −12.67755938927357, −12.40316635330535, −11.94372356967603, −11.40025099226867, −10.90308844947039, −10.29159121726664, −9.900728244815315, −9.391104160930403, −8.607682295570679, −8.073998334979667, −7.688434403986264, −6.989319474281580, −6.569404182809934, −5.861872849321662, −5.478684440182131, −4.789086546250320, −4.364452927617436, −3.738251753269743, −2.898935218702742, −2.642101473460416, −1.795103471130764, −1.015860633203888, 0,
1.015860633203888, 1.795103471130764, 2.642101473460416, 2.898935218702742, 3.738251753269743, 4.364452927617436, 4.789086546250320, 5.478684440182131, 5.861872849321662, 6.569404182809934, 6.989319474281580, 7.688434403986264, 8.073998334979667, 8.607682295570679, 9.391104160930403, 9.900728244815315, 10.29159121726664, 10.90308844947039, 11.40025099226867, 11.94372356967603, 12.40316635330535, 12.67755938927357, 13.55306398572525, 13.81199799204114, 14.09410436402507
