| L(s) = 1 | − 5·5-s + 34.6·7-s − 23.2·11-s + 60.0·13-s + 19.6·17-s + 10.2·19-s − 79.7·23-s + 25·25-s + 110.·29-s + 42.5·31-s − 173.·35-s + 308.·37-s − 106.·41-s + 467.·43-s − 37.7·47-s + 858.·49-s − 568.·53-s + 116.·55-s + 666.·59-s − 862.·61-s − 300.·65-s − 547.·67-s + 761.·71-s − 216.·73-s − 805.·77-s − 258.·79-s + 903.·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.87·7-s − 0.636·11-s + 1.28·13-s + 0.280·17-s + 0.123·19-s − 0.723·23-s + 0.200·25-s + 0.709·29-s + 0.246·31-s − 0.837·35-s + 1.37·37-s − 0.407·41-s + 1.65·43-s − 0.117·47-s + 2.50·49-s − 1.47·53-s + 0.284·55-s + 1.47·59-s − 1.80·61-s − 0.573·65-s − 0.997·67-s + 1.27·71-s − 0.347·73-s − 1.19·77-s − 0.368·79-s + 1.19·83-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
| \(L(2)\) |
\(\approx\) |
\(2.898623538\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.898623538\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 34.6T + 343T^{2} \) |
| 11 | \( 1 + 23.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 60.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 19.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 10.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 79.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 110.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 42.5T + 2.97e4T^{2} \) |
| 37 | \( 1 - 308.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 106.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 467.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 37.7T + 1.03e5T^{2} \) |
| 53 | \( 1 + 568.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 666.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 862.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 547.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 761.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 216.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 258.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 903.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.26e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 617.T + 9.12e5T^{2} \) |
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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
−8.420500656550136435031608209594, −8.029327566109462965661782598940, −7.48771721411008604669341244824, −6.27153002861706750575730209513, −5.48901015247237656875389974818, −4.62743202223805631419636825434, −4.01623971858255912500607427275, −2.80299950164098251830264048593, −1.68821790046771359615436816830, −0.825906194123945619465300611495,
0.825906194123945619465300611495, 1.68821790046771359615436816830, 2.80299950164098251830264048593, 4.01623971858255912500607427275, 4.62743202223805631419636825434, 5.48901015247237656875389974818, 6.27153002861706750575730209513, 7.48771721411008604669341244824, 8.029327566109462965661782598940, 8.420500656550136435031608209594
