| L(s) = 1 | − 3-s + 5-s + 9-s + 2·11-s + 4·13-s − 15-s + 6·17-s + 8·19-s + 23-s + 25-s − 27-s + 4·29-s − 2·33-s − 2·37-s − 4·39-s − 2·41-s − 2·43-s + 45-s + 12·47-s − 7·49-s − 6·51-s − 6·53-s + 2·55-s − 8·57-s − 8·59-s + 2·61-s + 4·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.603·11-s + 1.10·13-s − 0.258·15-s + 1.45·17-s + 1.83·19-s + 0.208·23-s + 1/5·25-s − 0.192·27-s + 0.742·29-s − 0.348·33-s − 0.328·37-s − 0.640·39-s − 0.312·41-s − 0.304·43-s + 0.149·45-s + 1.75·47-s − 49-s − 0.840·51-s − 0.824·53-s + 0.269·55-s − 1.05·57-s − 1.04·59-s + 0.256·61-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.344413734\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.344413734\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 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
−8.099829289251043056017865920156, −7.36774795875050200479729686513, −6.66016768918830084650333494318, −5.87162779404798123328649467192, −5.46774177718589897563459500213, −4.61153870419235948642789565496, −3.58619045821314009673705835228, −3.00826336935468801990164382087, −1.51322780352927555501790305391, −0.972194799436486776940325096095,
0.972194799436486776940325096095, 1.51322780352927555501790305391, 3.00826336935468801990164382087, 3.58619045821314009673705835228, 4.61153870419235948642789565496, 5.46774177718589897563459500213, 5.87162779404798123328649467192, 6.66016768918830084650333494318, 7.36774795875050200479729686513, 8.099829289251043056017865920156
