| L(s) = 1 | − 2-s + 4-s + 5-s − 4.45·7-s − 8-s − 10-s + 0.479·11-s + 5.00·13-s + 4.45·14-s + 16-s + 6.43·17-s + 5.04·19-s + 20-s − 0.479·22-s + 5.25·23-s + 25-s − 5.00·26-s − 4.45·28-s − 8.95·29-s + 7.90·31-s − 32-s − 6.43·34-s − 4.45·35-s − 6.18·37-s − 5.04·38-s − 40-s + 10.6·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.447·5-s − 1.68·7-s − 0.353·8-s − 0.316·10-s + 0.144·11-s + 1.38·13-s + 1.19·14-s + 0.250·16-s + 1.56·17-s + 1.15·19-s + 0.223·20-s − 0.102·22-s + 1.09·23-s + 0.200·25-s − 0.980·26-s − 0.842·28-s − 1.66·29-s + 1.42·31-s − 0.176·32-s − 1.10·34-s − 0.753·35-s − 1.01·37-s − 0.819·38-s − 0.158·40-s + 1.66·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.572705674\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.572705674\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 89 | \( 1 + T \) |
| good | 7 | \( 1 + 4.45T + 7T^{2} \) |
| 11 | \( 1 - 0.479T + 11T^{2} \) |
| 13 | \( 1 - 5.00T + 13T^{2} \) |
| 17 | \( 1 - 6.43T + 17T^{2} \) |
| 19 | \( 1 - 5.04T + 19T^{2} \) |
| 23 | \( 1 - 5.25T + 23T^{2} \) |
| 29 | \( 1 + 8.95T + 29T^{2} \) |
| 31 | \( 1 - 7.90T + 31T^{2} \) |
| 37 | \( 1 + 6.18T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 - 0.180T + 47T^{2} \) |
| 53 | \( 1 + 12.3T + 53T^{2} \) |
| 59 | \( 1 - 7.58T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 + 8.29T + 67T^{2} \) |
| 71 | \( 1 + 4.33T + 71T^{2} \) |
| 73 | \( 1 + 8.00T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 - 7.69T + 83T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−7.62076617967582599330704327836, −7.35090863698799733764078968129, −6.32699994069431082026269215811, −5.99838669285952601588860845123, −5.38613547881694240585538534448, −4.03502501492137636051674554141, −3.18110640895113091036840724024, −2.92271840642280773383654287730, −1.45810515866687204163083618767, −0.75414630101385622566077507069,
0.75414630101385622566077507069, 1.45810515866687204163083618767, 2.92271840642280773383654287730, 3.18110640895113091036840724024, 4.03502501492137636051674554141, 5.38613547881694240585538534448, 5.99838669285952601588860845123, 6.32699994069431082026269215811, 7.35090863698799733764078968129, 7.62076617967582599330704327836
