| L(s) = 1 | + 1.99·2-s + 1.97·4-s − 0.735·5-s − 3.83·7-s − 0.0540·8-s − 1.46·10-s − 0.205·11-s + 6.99·13-s − 7.64·14-s − 4.05·16-s − 3.70·17-s + 4.87·19-s − 1.45·20-s − 0.409·22-s − 23-s − 4.45·25-s + 13.9·26-s − 7.56·28-s + 29-s + 5.25·31-s − 7.97·32-s − 7.39·34-s + 2.81·35-s + 4.20·37-s + 9.72·38-s + 0.0396·40-s − 11.5·41-s + ⋯ |
| L(s) = 1 | + 1.40·2-s + 0.986·4-s − 0.328·5-s − 1.44·7-s − 0.0190·8-s − 0.463·10-s − 0.0619·11-s + 1.94·13-s − 2.04·14-s − 1.01·16-s − 0.899·17-s + 1.11·19-s − 0.324·20-s − 0.0873·22-s − 0.208·23-s − 0.891·25-s + 2.73·26-s − 1.42·28-s + 0.185·29-s + 0.943·31-s − 1.40·32-s − 1.26·34-s + 0.476·35-s + 0.691·37-s + 1.57·38-s + 0.00627·40-s − 1.80·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.169743414\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.169743414\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 1.99T + 2T^{2} \) |
| 5 | \( 1 + 0.735T + 5T^{2} \) |
| 7 | \( 1 + 3.83T + 7T^{2} \) |
| 11 | \( 1 + 0.205T + 11T^{2} \) |
| 13 | \( 1 - 6.99T + 13T^{2} \) |
| 17 | \( 1 + 3.70T + 17T^{2} \) |
| 19 | \( 1 - 4.87T + 19T^{2} \) |
| 31 | \( 1 - 5.25T + 31T^{2} \) |
| 37 | \( 1 - 4.20T + 37T^{2} \) |
| 41 | \( 1 + 11.5T + 41T^{2} \) |
| 43 | \( 1 - 6.44T + 43T^{2} \) |
| 47 | \( 1 - 0.322T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 - 8.24T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 - 7.87T + 67T^{2} \) |
| 71 | \( 1 - 8.34T + 71T^{2} \) |
| 73 | \( 1 - 1.72T + 73T^{2} \) |
| 79 | \( 1 + 2.27T + 79T^{2} \) |
| 83 | \( 1 - 2.31T + 83T^{2} \) |
| 89 | \( 1 - 9.00T + 89T^{2} \) |
| 97 | \( 1 - 15.7T + 97T^{2} \) |
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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
−8.060810262797284452839001994843, −6.84258143511583936439616644714, −6.58945162743928402538448684629, −5.84785216162353719158111611247, −5.31121142509836296472542245801, −4.14849001476748014849162381154, −3.76145821601839053555768766516, −3.17245104649901026901442826906, −2.29147163329020714682736537918, −0.73860061738478056951554600119,
0.73860061738478056951554600119, 2.29147163329020714682736537918, 3.17245104649901026901442826906, 3.76145821601839053555768766516, 4.14849001476748014849162381154, 5.31121142509836296472542245801, 5.84785216162353719158111611247, 6.58945162743928402538448684629, 6.84258143511583936439616644714, 8.060810262797284452839001994843
