L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s − 7-s + 8-s + 9-s − 10-s + 3·11-s + 12-s − 13-s − 14-s − 15-s + 16-s + 2·17-s + 18-s − 20-s − 21-s + 3·22-s − 23-s + 24-s + 25-s − 26-s + 27-s − 28-s − 30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.904·11-s + 0.288·12-s − 0.277·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.223·20-s − 0.218·21-s + 0.639·22-s − 0.208·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s − 0.188·28-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 29 | \( 1 \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 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
−12.91030092666140, −12.35293928308160, −12.12804283434467, −11.55245440583076, −11.03642229914351, −10.75836871235101, −9.913243784068702, −9.787658976123213, −9.071742886124835, −8.809502642526320, −8.164178087352048, −7.625724672659549, −7.315958397717068, −6.712995668478146, −6.446307763910263, −5.682488597263389, −5.316360944245793, −4.717197706055261, −4.084413183184721, −3.706686715846733, −3.395313201733350, −2.762851770662620, −2.116204321954916, −1.606179891436692, −0.8920089811016385, 0,
0.8920089811016385, 1.606179891436692, 2.116204321954916, 2.762851770662620, 3.395313201733350, 3.706686715846733, 4.084413183184721, 4.717197706055261, 5.316360944245793, 5.682488597263389, 6.446307763910263, 6.712995668478146, 7.315958397717068, 7.625724672659549, 8.164178087352048, 8.809502642526320, 9.071742886124835, 9.787658976123213, 9.913243784068702, 10.75836871235101, 11.03642229914351, 11.55245440583076, 12.12804283434467, 12.35293928308160, 12.91030092666140