| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s + 2·11-s − 12-s − 4·13-s + 16-s + 6·17-s − 18-s + 8·19-s − 2·22-s + 24-s + 4·26-s − 27-s + 4·29-s − 32-s − 2·33-s − 6·34-s + 36-s − 2·37-s − 8·38-s + 4·39-s − 2·41-s + 2·43-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 + 0.603·11-s − 0.288·12-s − 1.10·13-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 1.83·19-s − 0.426·22-s + 0.204·24-s + 0.784·26-s − 0.192·27-s + 0.742·29-s − 0.176·32-s − 0.348·33-s − 1.02·34-s + 1/6·36-s − 0.328·37-s − 1.29·38-s + 0.640·39-s − 0.312·41-s + 0.304·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 79350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.748941664\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.748941664\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−14.02894136355891, −13.67811391216491, −12.72006601635359, −12.31935617972809, −11.93094529176418, −11.64855955776296, −10.98274493329250, −10.37124791146626, −9.969492304550881, −9.516546038149684, −9.189045395510213, −8.389421084026768, −7.688592543455992, −7.525816575942572, −6.900313765511577, −6.352338172021143, −5.692519195380841, −5.199393954577609, −4.772723297699937, −3.824833589910487, −3.279727092732983, −2.658270438364894, −1.807736634466505, −1.063217639494199, −0.6050343173520769,
0.6050343173520769, 1.063217639494199, 1.807736634466505, 2.658270438364894, 3.279727092732983, 3.824833589910487, 4.772723297699937, 5.199393954577609, 5.692519195380841, 6.352338172021143, 6.900313765511577, 7.525816575942572, 7.688592543455992, 8.389421084026768, 9.189045395510213, 9.516546038149684, 9.969492304550881, 10.37124791146626, 10.98274493329250, 11.64855955776296, 11.93094529176418, 12.31935617972809, 12.72006601635359, 13.67811391216491, 14.02894136355891
