| L(s) = 1 | − 0.568·2-s + 0.674·3-s − 1.67·4-s + 5-s − 0.383·6-s + 2.09·8-s − 2.54·9-s − 0.568·10-s − 11-s − 1.13·12-s + 1.13·13-s + 0.674·15-s + 2.16·16-s − 0.503·17-s + 1.44·18-s − 5.40·19-s − 1.67·20-s + 0.568·22-s + 0.365·23-s + 1.40·24-s + 25-s − 0.644·26-s − 3.73·27-s + 7.02·29-s − 0.383·30-s + 5.91·31-s − 5.41·32-s + ⋯ |
| L(s) = 1 | − 0.401·2-s + 0.389·3-s − 0.838·4-s + 0.447·5-s − 0.156·6-s + 0.738·8-s − 0.848·9-s − 0.179·10-s − 0.301·11-s − 0.326·12-s + 0.314·13-s + 0.174·15-s + 0.541·16-s − 0.122·17-s + 0.341·18-s − 1.24·19-s − 0.374·20-s + 0.121·22-s + 0.0763·23-s + 0.287·24-s + 0.200·25-s − 0.126·26-s − 0.719·27-s + 1.30·29-s − 0.0699·30-s + 1.06·31-s − 0.956·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.214037731\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.214037731\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 0.568T + 2T^{2} \) |
| 3 | \( 1 - 0.674T + 3T^{2} \) |
| 13 | \( 1 - 1.13T + 13T^{2} \) |
| 17 | \( 1 + 0.503T + 17T^{2} \) |
| 19 | \( 1 + 5.40T + 19T^{2} \) |
| 23 | \( 1 - 0.365T + 23T^{2} \) |
| 29 | \( 1 - 7.02T + 29T^{2} \) |
| 31 | \( 1 - 5.91T + 31T^{2} \) |
| 37 | \( 1 + 1.70T + 37T^{2} \) |
| 41 | \( 1 + 1.38T + 41T^{2} \) |
| 43 | \( 1 - 9.60T + 43T^{2} \) |
| 47 | \( 1 - 1.01T + 47T^{2} \) |
| 53 | \( 1 + 0.0335T + 53T^{2} \) |
| 59 | \( 1 - 6.99T + 59T^{2} \) |
| 61 | \( 1 - 4.73T + 61T^{2} \) |
| 67 | \( 1 + 7.09T + 67T^{2} \) |
| 71 | \( 1 - 1.01T + 71T^{2} \) |
| 73 | \( 1 + 8.82T + 73T^{2} \) |
| 79 | \( 1 - 6.44T + 79T^{2} \) |
| 83 | \( 1 - 4.08T + 83T^{2} \) |
| 89 | \( 1 - 2.62T + 89T^{2} \) |
| 97 | \( 1 - 17.6T + 97T^{2} \) |
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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
−8.684373914983113697429135098772, −8.425283439164761158184389742654, −7.59389074433089895323821445966, −6.48921639275951774415247993438, −5.77960709718406558753205347868, −4.87354122040174085720223292392, −4.12130712868277734807983449755, −3.04240513770321544768919213513, −2.11520089283046760080122197333, −0.71546895242009699472749332937,
0.71546895242009699472749332937, 2.11520089283046760080122197333, 3.04240513770321544768919213513, 4.12130712868277734807983449755, 4.87354122040174085720223292392, 5.77960709718406558753205347868, 6.48921639275951774415247993438, 7.59389074433089895323821445966, 8.425283439164761158184389742654, 8.684373914983113697429135098772
