| L(s) = 1 | − 1.90·3-s − 5-s − 7-s + 0.631·9-s + 6.09·11-s + 13-s + 1.90·15-s + 7.44·17-s + 0.655·19-s + 1.90·21-s − 4.09·23-s + 25-s + 4.51·27-s − 8.00·29-s − 1.21·31-s − 11.6·33-s + 35-s − 1.75·37-s − 1.90·39-s − 8.81·41-s + 2.28·43-s − 0.631·45-s + 5.12·47-s + 49-s − 14.1·51-s − 1.12·53-s − 6.09·55-s + ⋯ |
| L(s) = 1 | − 1.10·3-s − 0.447·5-s − 0.377·7-s + 0.210·9-s + 1.83·11-s + 0.277·13-s + 0.492·15-s + 1.80·17-s + 0.150·19-s + 0.415·21-s − 0.854·23-s + 0.200·25-s + 0.868·27-s − 1.48·29-s − 0.218·31-s − 2.02·33-s + 0.169·35-s − 0.289·37-s − 0.305·39-s − 1.37·41-s + 0.348·43-s − 0.0941·45-s + 0.747·47-s + 0.142·49-s − 1.98·51-s − 0.154·53-s − 0.822·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.108342776\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.108342776\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 1.90T + 3T^{2} \) |
| 11 | \( 1 - 6.09T + 11T^{2} \) |
| 17 | \( 1 - 7.44T + 17T^{2} \) |
| 19 | \( 1 - 0.655T + 19T^{2} \) |
| 23 | \( 1 + 4.09T + 23T^{2} \) |
| 29 | \( 1 + 8.00T + 29T^{2} \) |
| 31 | \( 1 + 1.21T + 31T^{2} \) |
| 37 | \( 1 + 1.75T + 37T^{2} \) |
| 41 | \( 1 + 8.81T + 41T^{2} \) |
| 43 | \( 1 - 2.28T + 43T^{2} \) |
| 47 | \( 1 - 5.12T + 47T^{2} \) |
| 53 | \( 1 + 1.12T + 53T^{2} \) |
| 59 | \( 1 - 1.51T + 59T^{2} \) |
| 61 | \( 1 + 8.24T + 61T^{2} \) |
| 67 | \( 1 - 15.2T + 67T^{2} \) |
| 71 | \( 1 + 15.0T + 71T^{2} \) |
| 73 | \( 1 + 4.49T + 73T^{2} \) |
| 79 | \( 1 - 2.41T + 79T^{2} \) |
| 83 | \( 1 - 3.74T + 83T^{2} \) |
| 89 | \( 1 - 16.1T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−8.602030976640632600575055491058, −7.63263920173413915535502190352, −6.98046165376007469925203194450, −6.11421904193602056197510330097, −5.77817203900785957015262716269, −4.82759722654380059826067951943, −3.78852367788728738351178154260, −3.37498204257911760800811433131, −1.67840814059649002135743347666, −0.68162738238628253825191714430,
0.68162738238628253825191714430, 1.67840814059649002135743347666, 3.37498204257911760800811433131, 3.78852367788728738351178154260, 4.82759722654380059826067951943, 5.77817203900785957015262716269, 6.11421904193602056197510330097, 6.98046165376007469925203194450, 7.63263920173413915535502190352, 8.602030976640632600575055491058
