L(s) = 1 | − 3·3-s − 0.726·5-s + 9·9-s − 64.4·11-s − 71.8·13-s + 2.17·15-s − 48.9·17-s − 34.3·19-s − 0.903·23-s − 124.·25-s − 27·27-s + 226.·29-s + 275.·31-s + 193.·33-s + 295.·37-s + 215.·39-s − 186.·41-s − 455.·43-s − 6.53·45-s − 282.·47-s + 146.·51-s + 356.·53-s + 46.8·55-s + 103.·57-s − 729.·59-s − 274.·61-s + 52.1·65-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.0649·5-s + 0.333·9-s − 1.76·11-s − 1.53·13-s + 0.0375·15-s − 0.697·17-s − 0.415·19-s − 0.00818·23-s − 0.995·25-s − 0.192·27-s + 1.45·29-s + 1.59·31-s + 1.02·33-s + 1.31·37-s + 0.884·39-s − 0.710·41-s − 1.61·43-s − 0.0216·45-s − 0.876·47-s + 0.402·51-s + 0.923·53-s + 0.114·55-s + 0.239·57-s − 1.60·59-s − 0.575·61-s + 0.0995·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.6774938506\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6774938506\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 0.726T + 125T^{2} \) |
| 11 | \( 1 + 64.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 71.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 48.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 34.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 0.903T + 1.21e4T^{2} \) |
| 29 | \( 1 - 226.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 275.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 295.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 186.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 455.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 282.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 356.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 729.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 274.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 193.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 40.5T + 3.57e5T^{2} \) |
| 73 | \( 1 - 206.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 937.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 911.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 949.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 39.4T + 9.12e5T^{2} \) |
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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
−9.742886783470216858716401104069, −8.344805000675161999752020638601, −7.81819344948203337852503282905, −6.87197922816662345814983150936, −6.03863845004276734067227566078, −4.91476947393261707548840790013, −4.61762761760973827952684631634, −2.95868675910434276037013723042, −2.14213656842058226812094899533, −0.40984204004881369909836243784,
0.40984204004881369909836243784, 2.14213656842058226812094899533, 2.95868675910434276037013723042, 4.61762761760973827952684631634, 4.91476947393261707548840790013, 6.03863845004276734067227566078, 6.87197922816662345814983150936, 7.81819344948203337852503282905, 8.344805000675161999752020638601, 9.742886783470216858716401104069