| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 3·9-s − 4·11-s + 13-s − 14-s + 16-s + 3·17-s − 3·18-s + 8·19-s − 4·22-s − 23-s + 26-s − 28-s − 8·29-s + 4·31-s + 32-s + 3·34-s − 3·36-s − 5·37-s + 8·38-s + 6·41-s − 2·43-s − 4·44-s − 46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 9-s − 1.20·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.707·18-s + 1.83·19-s − 0.852·22-s − 0.208·23-s + 0.196·26-s − 0.188·28-s − 1.48·29-s + 0.718·31-s + 0.176·32-s + 0.514·34-s − 1/2·36-s − 0.821·37-s + 1.29·38-s + 0.937·41-s − 0.304·43-s − 0.603·44-s − 0.147·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 7 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 17 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−7.67464733429060645778310131267, −6.69872776615220740694218802442, −5.73728690380499158817190403585, −5.56193316633919703115422799526, −4.83876383606479452472076041801, −3.73559610665724590884101397729, −3.10338747966418723612624547974, −2.59745316070575425828374178212, −1.36643612662942117847805433889, 0,
1.36643612662942117847805433889, 2.59745316070575425828374178212, 3.10338747966418723612624547974, 3.73559610665724590884101397729, 4.83876383606479452472076041801, 5.56193316633919703115422799526, 5.73728690380499158817190403585, 6.69872776615220740694218802442, 7.67464733429060645778310131267
