| L(s) = 1 | − 5·5-s − 9.46·7-s + 49.0·11-s + 55.4·13-s + 27.9·17-s − 26.3·19-s + 9.53·23-s + 25·25-s − 218.·29-s + 158.·31-s + 47.3·35-s + 189.·37-s + 246.·41-s − 40.2·43-s − 213.·47-s − 253.·49-s + 283.·53-s − 245.·55-s − 92·59-s + 370.·61-s − 277.·65-s − 1.08e3·67-s + 333.·71-s − 606.·73-s − 464.·77-s + 44.3·79-s − 107.·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.511·7-s + 1.34·11-s + 1.18·13-s + 0.399·17-s − 0.318·19-s + 0.0864·23-s + 0.200·25-s − 1.39·29-s + 0.920·31-s + 0.228·35-s + 0.842·37-s + 0.937·41-s − 0.142·43-s − 0.664·47-s − 0.738·49-s + 0.734·53-s − 0.601·55-s − 0.203·59-s + 0.776·61-s − 0.528·65-s − 1.98·67-s + 0.557·71-s − 0.972·73-s − 0.687·77-s + 0.0631·79-s − 0.142·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.153205989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.153205989\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| good | 7 | \( 1 + 9.46T + 343T^{2} \) |
| 11 | \( 1 - 49.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 55.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 27.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 26.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 9.53T + 1.21e4T^{2} \) |
| 29 | \( 1 + 218.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 158.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 189.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 246.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 40.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 213.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 283.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 92T + 2.05e5T^{2} \) |
| 61 | \( 1 - 370.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.08e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 333.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 606.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 44.3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 107.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 257.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.31e3T + 9.12e5T^{2} \) |
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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
−8.782936960327829760281668685780, −7.995293022816128128620996962316, −7.11628665538161078952480085531, −6.32682588842465689425818093719, −5.79449024619700863114740012463, −4.46595466996278286007341673799, −3.81194870470042003794609257339, −3.06807169338575925706661682475, −1.65872956166561162131142991731, −0.69715916994554157241350064901,
0.69715916994554157241350064901, 1.65872956166561162131142991731, 3.06807169338575925706661682475, 3.81194870470042003794609257339, 4.46595466996278286007341673799, 5.79449024619700863114740012463, 6.32682588842465689425818093719, 7.11628665538161078952480085531, 7.995293022816128128620996962316, 8.782936960327829760281668685780
