L(s) = 1 | + 2-s + 4-s − 5-s + 7-s + 8-s − 3·9-s − 10-s − 4·11-s + 6·13-s + 14-s + 16-s − 2·17-s − 3·18-s − 20-s − 4·22-s + 25-s + 6·26-s + 28-s − 6·29-s + 8·31-s + 32-s − 2·34-s − 35-s − 3·36-s − 10·37-s − 40-s + 4·43-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s + 0.353·8-s − 9-s − 0.316·10-s − 1.20·11-s + 1.66·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.707·18-s − 0.223·20-s − 0.852·22-s + 1/5·25-s + 1.17·26-s + 0.188·28-s − 1.11·29-s + 1.43·31-s + 0.176·32-s − 0.342·34-s − 0.169·35-s − 1/2·36-s − 1.64·37-s − 0.158·40-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 41 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.85794115542236, −13.45665288389724, −12.90899083024432, −12.41067883787437, −11.92039794704288, −11.26462214063963, −11.08495594034025, −10.70536900059630, −10.14316254108667, −9.335023571857829, −8.743700751832752, −8.349533429581089, −7.911024787044637, −7.489902343542224, −6.612807662961658, −6.294091909268294, −5.725107415735820, −5.122327915311194, −4.837018755556024, −4.015150524644884, −3.487370214172761, −3.084598707452445, −2.350156223280730, −1.775738475158031, −0.8631895797776376, 0,
0.8631895797776376, 1.775738475158031, 2.350156223280730, 3.084598707452445, 3.487370214172761, 4.015150524644884, 4.837018755556024, 5.122327915311194, 5.725107415735820, 6.294091909268294, 6.612807662961658, 7.489902343542224, 7.911024787044637, 8.349533429581089, 8.743700751832752, 9.335023571857829, 10.14316254108667, 10.70536900059630, 11.08495594034025, 11.26462214063963, 11.92039794704288, 12.41067883787437, 12.90899083024432, 13.45665288389724, 13.85794115542236