| L(s) = 1 | + 1.59·2-s + 2.79·3-s + 0.557·4-s − 0.652·5-s + 4.46·6-s − 2.30·8-s + 4.80·9-s − 1.04·10-s − 1.63·11-s + 1.55·12-s + 3.69·13-s − 1.82·15-s − 4.80·16-s + 7.52·17-s + 7.67·18-s + 1.61·19-s − 0.364·20-s − 2.61·22-s + 5.52·23-s − 6.44·24-s − 4.57·25-s + 5.90·26-s + 5.02·27-s − 3.79·29-s − 2.91·30-s + 2.77·31-s − 3.07·32-s + ⋯ |
| L(s) = 1 | + 1.13·2-s + 1.61·3-s + 0.278·4-s − 0.291·5-s + 1.82·6-s − 0.815·8-s + 1.60·9-s − 0.330·10-s − 0.492·11-s + 0.449·12-s + 1.02·13-s − 0.470·15-s − 1.20·16-s + 1.82·17-s + 1.80·18-s + 0.369·19-s − 0.0814·20-s − 0.556·22-s + 1.15·23-s − 1.31·24-s − 0.914·25-s + 1.15·26-s + 0.967·27-s − 0.704·29-s − 0.532·30-s + 0.497·31-s − 0.542·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.653875344\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.653875344\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 - 1.59T + 2T^{2} \) |
| 3 | \( 1 - 2.79T + 3T^{2} \) |
| 5 | \( 1 + 0.652T + 5T^{2} \) |
| 11 | \( 1 + 1.63T + 11T^{2} \) |
| 13 | \( 1 - 3.69T + 13T^{2} \) |
| 17 | \( 1 - 7.52T + 17T^{2} \) |
| 19 | \( 1 - 1.61T + 19T^{2} \) |
| 23 | \( 1 - 5.52T + 23T^{2} \) |
| 29 | \( 1 + 3.79T + 29T^{2} \) |
| 31 | \( 1 - 2.77T + 31T^{2} \) |
| 37 | \( 1 - 2.70T + 37T^{2} \) |
| 41 | \( 1 - 12.1T + 41T^{2} \) |
| 43 | \( 1 - 8.54T + 43T^{2} \) |
| 47 | \( 1 - 3.28T + 47T^{2} \) |
| 53 | \( 1 - 5.38T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 + 6.34T + 61T^{2} \) |
| 67 | \( 1 - 0.279T + 67T^{2} \) |
| 71 | \( 1 + 5.76T + 71T^{2} \) |
| 79 | \( 1 + 7.97T + 79T^{2} \) |
| 83 | \( 1 + 16.3T + 83T^{2} \) |
| 89 | \( 1 - 11.2T + 89T^{2} \) |
| 97 | \( 1 + 12.0T + 97T^{2} \) |
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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
−8.538870164616733799902071658403, −7.70492952550894977672946052682, −7.39730099380998909641527219830, −6.02472722241911054236919548438, −5.53844380203338188033994931515, −4.40046176685090266373698921486, −3.81738388727099902718786010514, −3.12988728443960306542444889488, −2.61208536666386928191440041548, −1.19781139667864648592626452555,
1.19781139667864648592626452555, 2.61208536666386928191440041548, 3.12988728443960306542444889488, 3.81738388727099902718786010514, 4.40046176685090266373698921486, 5.53844380203338188033994931515, 6.02472722241911054236919548438, 7.39730099380998909641527219830, 7.70492952550894977672946052682, 8.538870164616733799902071658403
