| L(s) = 1 | + 3-s + 3.48·5-s − 4.12·7-s + 9-s + 3.77·11-s − 1.77·13-s + 3.48·15-s + 1.43·17-s + 5.84·19-s − 4.12·21-s + 3.55·23-s + 7.12·25-s + 27-s + 7.03·29-s − 9.72·31-s + 3.77·33-s − 14.3·35-s + 1.08·37-s − 1.77·39-s + 9.67·41-s − 2.61·43-s + 3.48·45-s − 8.99·47-s + 10.0·49-s + 1.43·51-s − 7.25·53-s + 13.1·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.55·5-s − 1.55·7-s + 0.333·9-s + 1.13·11-s − 0.492·13-s + 0.899·15-s + 0.348·17-s + 1.34·19-s − 0.900·21-s + 0.741·23-s + 1.42·25-s + 0.192·27-s + 1.30·29-s − 1.74·31-s + 0.657·33-s − 2.42·35-s + 0.179·37-s − 0.284·39-s + 1.51·41-s − 0.398·43-s + 0.519·45-s − 1.31·47-s + 1.43·49-s + 0.201·51-s − 0.997·53-s + 1.77·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.260700253\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.260700253\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 67 | \( 1 - T \) |
| good | 5 | \( 1 - 3.48T + 5T^{2} \) |
| 7 | \( 1 + 4.12T + 7T^{2} \) |
| 11 | \( 1 - 3.77T + 11T^{2} \) |
| 13 | \( 1 + 1.77T + 13T^{2} \) |
| 17 | \( 1 - 1.43T + 17T^{2} \) |
| 19 | \( 1 - 5.84T + 19T^{2} \) |
| 23 | \( 1 - 3.55T + 23T^{2} \) |
| 29 | \( 1 - 7.03T + 29T^{2} \) |
| 31 | \( 1 + 9.72T + 31T^{2} \) |
| 37 | \( 1 - 1.08T + 37T^{2} \) |
| 41 | \( 1 - 9.67T + 41T^{2} \) |
| 43 | \( 1 + 2.61T + 43T^{2} \) |
| 47 | \( 1 + 8.99T + 47T^{2} \) |
| 53 | \( 1 + 7.25T + 53T^{2} \) |
| 59 | \( 1 + 14.2T + 59T^{2} \) |
| 61 | \( 1 + 8.88T + 61T^{2} \) |
| 71 | \( 1 + 2.22T + 71T^{2} \) |
| 73 | \( 1 - 4.27T + 73T^{2} \) |
| 79 | \( 1 - 15.2T + 79T^{2} \) |
| 83 | \( 1 - 0.669T + 83T^{2} \) |
| 89 | \( 1 + 5.52T + 89T^{2} \) |
| 97 | \( 1 - 9.88T + 97T^{2} \) |
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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
−9.846652816850481117209598754274, −9.415595829010832505279660685459, −9.101203638076036016923025942926, −7.55862840354414265438129592154, −6.61932027475933349176050700817, −6.08262397572684075613798284197, −4.99386838743371295420574894732, −3.48679632023936185954860294140, −2.75325141198962949383988410714, −1.38857101931261348512828429348,
1.38857101931261348512828429348, 2.75325141198962949383988410714, 3.48679632023936185954860294140, 4.99386838743371295420574894732, 6.08262397572684075613798284197, 6.61932027475933349176050700817, 7.55862840354414265438129592154, 9.101203638076036016923025942926, 9.415595829010832505279660685459, 9.846652816850481117209598754274
