| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s + 5·5-s + 6·6-s + 8·8-s + 9·9-s + 10·10-s − 4·11-s + 12·12-s + 42·13-s + 15·15-s + 16·16-s + 86·17-s + 18·18-s + 96·19-s + 20·20-s − 8·22-s − 96·23-s + 24·24-s + 25·25-s + 84·26-s + 27·27-s − 78·29-s + 30·30-s − 80·31-s + 32·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.109·11-s + 0.288·12-s + 0.896·13-s + 0.258·15-s + 1/4·16-s + 1.22·17-s + 0.235·18-s + 1.15·19-s + 0.223·20-s − 0.0775·22-s − 0.870·23-s + 0.204·24-s + 1/5·25-s + 0.633·26-s + 0.192·27-s − 0.499·29-s + 0.182·30-s − 0.463·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.492782627\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.492782627\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - p T \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p^{3} T^{2} \) |
| 13 | \( 1 - 42 T + p^{3} T^{2} \) |
| 17 | \( 1 - 86 T + p^{3} T^{2} \) |
| 19 | \( 1 - 96 T + p^{3} T^{2} \) |
| 23 | \( 1 + 96 T + p^{3} T^{2} \) |
| 29 | \( 1 + 78 T + p^{3} T^{2} \) |
| 31 | \( 1 + 80 T + p^{3} T^{2} \) |
| 37 | \( 1 - 50 T + p^{3} T^{2} \) |
| 41 | \( 1 - 26 T + p^{3} T^{2} \) |
| 43 | \( 1 + 32 T + p^{3} T^{2} \) |
| 47 | \( 1 - 20 T + p^{3} T^{2} \) |
| 53 | \( 1 + 382 T + p^{3} T^{2} \) |
| 59 | \( 1 + 356 T + p^{3} T^{2} \) |
| 61 | \( 1 - 134 T + p^{3} T^{2} \) |
| 67 | \( 1 - 888 T + p^{3} T^{2} \) |
| 71 | \( 1 - 868 T + p^{3} T^{2} \) |
| 73 | \( 1 - 70 T + p^{3} T^{2} \) |
| 79 | \( 1 - 400 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1052 T + p^{3} T^{2} \) |
| 89 | \( 1 - 634 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1202 T + p^{3} T^{2} \) |
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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
−9.277053975596413924145967733582, −8.134097703355078851570694685449, −7.61785780079026545693710206632, −6.56827002870660672784966910446, −5.75038444347863983293025254992, −5.04869577536359144245331291721, −3.80287220558949778162613857186, −3.24483844668393776200885018660, −2.09510919109010912703813405110, −1.08439013248129115798131899711,
1.08439013248129115798131899711, 2.09510919109010912703813405110, 3.24483844668393776200885018660, 3.80287220558949778162613857186, 5.04869577536359144245331291721, 5.75038444347863983293025254992, 6.56827002870660672784966910446, 7.61785780079026545693710206632, 8.134097703355078851570694685449, 9.277053975596413924145967733582
