L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 6·11-s − 12-s + 6·13-s + 16-s − 18-s − 6·19-s + 6·22-s − 23-s + 24-s − 6·26-s − 27-s − 6·29-s − 2·31-s − 32-s + 6·33-s + 36-s + 4·37-s + 6·38-s − 6·39-s − 6·41-s + 8·43-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 1.80·11-s − 0.288·12-s + 1.66·13-s + 1/4·16-s − 0.235·18-s − 1.37·19-s + 1.27·22-s − 0.208·23-s + 0.204·24-s − 1.17·26-s − 0.192·27-s − 1.11·29-s − 0.359·31-s − 0.176·32-s + 1.04·33-s + 1/6·36-s + 0.657·37-s + 0.973·38-s − 0.960·39-s − 0.937·41-s + 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 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
−13.16903368654808, −13.02450430055310, −12.69420859925302, −11.75541792996670, −11.53632765495532, −10.92516175185624, −10.52833758415922, −10.41285086850071, −9.741508112405402, −9.075818936713450, −8.646804070850322, −8.136858229778367, −7.850245551805027, −7.125614517646876, −6.757816001308880, −6.000002777863368, −5.728356209227875, −5.341302093607710, −4.386811068462805, −4.119812794435376, −3.236900037120018, −2.762870344232698, −1.945463615995204, −1.557454507772930, −0.5975219163571643, 0,
0.5975219163571643, 1.557454507772930, 1.945463615995204, 2.762870344232698, 3.236900037120018, 4.119812794435376, 4.386811068462805, 5.341302093607710, 5.728356209227875, 6.000002777863368, 6.757816001308880, 7.125614517646876, 7.850245551805027, 8.136858229778367, 8.646804070850322, 9.075818936713450, 9.741508112405402, 10.41285086850071, 10.52833758415922, 10.92516175185624, 11.53632765495532, 11.75541792996670, 12.69420859925302, 13.02450430055310, 13.16903368654808