| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s + 6·5-s + 6·6-s − 8·8-s + 9·9-s − 12·10-s − 30·11-s − 12·12-s − 53·13-s − 18·15-s + 16·16-s + 84·17-s − 18·18-s + 97·19-s + 24·20-s + 60·22-s + 84·23-s + 24·24-s − 89·25-s + 106·26-s − 27·27-s − 180·29-s + 36·30-s − 179·31-s − 32·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.536·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.379·10-s − 0.822·11-s − 0.288·12-s − 1.13·13-s − 0.309·15-s + 1/4·16-s + 1.19·17-s − 0.235·18-s + 1.17·19-s + 0.268·20-s + 0.581·22-s + 0.761·23-s + 0.204·24-s − 0.711·25-s + 0.799·26-s − 0.192·27-s − 1.15·29-s + 0.219·30-s − 1.03·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 + p T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 6 T + p^{3} T^{2} \) |
| 11 | \( 1 + 30 T + p^{3} T^{2} \) |
| 13 | \( 1 + 53 T + p^{3} T^{2} \) |
| 17 | \( 1 - 84 T + p^{3} T^{2} \) |
| 19 | \( 1 - 97 T + p^{3} T^{2} \) |
| 23 | \( 1 - 84 T + p^{3} T^{2} \) |
| 29 | \( 1 + 180 T + p^{3} T^{2} \) |
| 31 | \( 1 + 179 T + p^{3} T^{2} \) |
| 37 | \( 1 + 145 T + p^{3} T^{2} \) |
| 41 | \( 1 + 126 T + p^{3} T^{2} \) |
| 43 | \( 1 + 325 T + p^{3} T^{2} \) |
| 47 | \( 1 - 366 T + p^{3} T^{2} \) |
| 53 | \( 1 + 768 T + p^{3} T^{2} \) |
| 59 | \( 1 - 264 T + p^{3} T^{2} \) |
| 61 | \( 1 + 818 T + p^{3} T^{2} \) |
| 67 | \( 1 + 523 T + p^{3} T^{2} \) |
| 71 | \( 1 + 342 T + p^{3} T^{2} \) |
| 73 | \( 1 - 43 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1171 T + p^{3} T^{2} \) |
| 83 | \( 1 - 810 T + p^{3} T^{2} \) |
| 89 | \( 1 - 600 T + p^{3} T^{2} \) |
| 97 | \( 1 + 386 T + p^{3} T^{2} \) |
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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
−10.68854085458426173530746153853, −9.905885013745740233549500040094, −9.269611931425399334853358582860, −7.76429942905638468196366247814, −7.20656661708284496687948559649, −5.76723377306686705929444114951, −5.10466640891820727483970198370, −3.14682476160132402511301727249, −1.65052023018136325975449749462, 0,
1.65052023018136325975449749462, 3.14682476160132402511301727249, 5.10466640891820727483970198370, 5.76723377306686705929444114951, 7.20656661708284496687948559649, 7.76429942905638468196366247814, 9.269611931425399334853358582860, 9.905885013745740233549500040094, 10.68854085458426173530746153853
