| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s − 2.58·11-s + 12-s − 2.82·13-s + 16-s − 1.17·17-s + 18-s − 19-s − 2.58·22-s − 4.24·23-s + 24-s − 5·25-s − 2.82·26-s + 27-s − 8·29-s + 3.65·31-s + 32-s − 2.58·33-s − 1.17·34-s + 36-s − 2·37-s − 38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.779·11-s + 0.288·12-s − 0.784·13-s + 0.250·16-s − 0.284·17-s + 0.235·18-s − 0.229·19-s − 0.551·22-s − 0.884·23-s + 0.204·24-s − 25-s − 0.554·26-s + 0.192·27-s − 1.48·29-s + 0.656·31-s + 0.176·32-s − 0.450·33-s − 0.200·34-s + 0.166·36-s − 0.328·37-s − 0.162·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 2.58T + 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 + 1.17T + 17T^{2} \) |
| 23 | \( 1 + 4.24T + 23T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 - 3.65T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 2.58T + 41T^{2} \) |
| 43 | \( 1 + 6.82T + 43T^{2} \) |
| 47 | \( 1 + 7.65T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 2.58T + 61T^{2} \) |
| 67 | \( 1 + 4.24T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 - 9.41T + 73T^{2} \) |
| 79 | \( 1 - 5.89T + 79T^{2} \) |
| 83 | \( 1 + 9.65T + 83T^{2} \) |
| 89 | \( 1 - 2.58T + 89T^{2} \) |
| 97 | \( 1 - 14.8T + 97T^{2} \) |
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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
−7.921001929553059299419863351635, −7.04685995207511844410217226451, −6.36729272800849629815598149622, −5.46607134147724155827063240056, −4.88143063728245082344418850699, −4.01371535972840415521766401067, −3.33792807182435153078022926359, −2.38293855217422406941284432673, −1.81268045784839536802099907160, 0,
1.81268045784839536802099907160, 2.38293855217422406941284432673, 3.33792807182435153078022926359, 4.01371535972840415521766401067, 4.88143063728245082344418850699, 5.46607134147724155827063240056, 6.36729272800849629815598149622, 7.04685995207511844410217226451, 7.921001929553059299419863351635
