| L(s) = 1 | − 2·4-s + 2.95·5-s + 1.80·7-s + 1.40·11-s + 1.56·13-s + 4·16-s + 4.88·17-s + 5.65·19-s − 5.91·20-s − 23-s + 3.75·25-s − 3.60·28-s − 29-s + 7.71·31-s + 5.32·35-s + 7.59·37-s + 7.12·41-s − 8.16·43-s − 2.81·44-s − 2.95·47-s − 3.75·49-s − 3.13·52-s − 7.51·53-s + 4.16·55-s + 2.88·59-s − 10.4·61-s − 8·64-s + ⋯ |
| L(s) = 1 | − 4-s + 1.32·5-s + 0.680·7-s + 0.424·11-s + 0.434·13-s + 16-s + 1.18·17-s + 1.29·19-s − 1.32·20-s − 0.208·23-s + 0.751·25-s − 0.680·28-s − 0.185·29-s + 1.38·31-s + 0.900·35-s + 1.24·37-s + 1.11·41-s − 1.24·43-s − 0.424·44-s − 0.431·47-s − 0.537·49-s − 0.434·52-s − 1.03·53-s + 0.562·55-s + 0.375·59-s − 1.34·61-s − 64-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
| \(L(1)\) |
\(\approx\) |
\(2.738995061\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.738995061\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2T^{2} \) |
| 5 | \( 1 - 2.95T + 5T^{2} \) |
| 7 | \( 1 - 1.80T + 7T^{2} \) |
| 11 | \( 1 - 1.40T + 11T^{2} \) |
| 13 | \( 1 - 1.56T + 13T^{2} \) |
| 17 | \( 1 - 4.88T + 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 31 | \( 1 - 7.71T + 31T^{2} \) |
| 37 | \( 1 - 7.59T + 37T^{2} \) |
| 41 | \( 1 - 7.12T + 41T^{2} \) |
| 43 | \( 1 + 8.16T + 43T^{2} \) |
| 47 | \( 1 + 2.95T + 47T^{2} \) |
| 53 | \( 1 + 7.51T + 53T^{2} \) |
| 59 | \( 1 - 2.88T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 + 3.60T + 67T^{2} \) |
| 71 | \( 1 - 1.79T + 71T^{2} \) |
| 73 | \( 1 + 7.80T + 73T^{2} \) |
| 79 | \( 1 - 9.17T + 79T^{2} \) |
| 83 | \( 1 + 1.74T + 83T^{2} \) |
| 89 | \( 1 - 15.9T + 89T^{2} \) |
| 97 | \( 1 - 1.35T + 97T^{2} \) |
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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
−7.976403165797613479766791486515, −7.69914547568021626364214677260, −6.35618987665363709636017615337, −5.93019991047864085412857165263, −5.15078009086232559601775520485, −4.67573538329511936932759906448, −3.64343657630046690546442878745, −2.84153815633090194194589725730, −1.57099340606459063245022611898, −1.00612361074723869498029590764,
1.00612361074723869498029590764, 1.57099340606459063245022611898, 2.84153815633090194194589725730, 3.64343657630046690546442878745, 4.67573538329511936932759906448, 5.15078009086232559601775520485, 5.93019991047864085412857165263, 6.35618987665363709636017615337, 7.69914547568021626364214677260, 7.976403165797613479766791486515
