| L(s) = 1 | + 5-s + 0.867·7-s − 4.08·11-s + 1.21·13-s + 7.82·17-s − 4.51·19-s + 2.86·23-s + 25-s + 6.29·29-s − 2.52·31-s + 0.867·35-s − 0.523·37-s − 8.12·41-s + 3.82·43-s − 1.39·47-s − 6.24·49-s + 13.9·53-s − 4.08·55-s + 6.87·59-s + 9.08·61-s + 1.21·65-s + 3.37·67-s + 3.21·71-s + 8.60·73-s − 3.54·77-s + 16.2·79-s + 6.44·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.327·7-s − 1.23·11-s + 0.336·13-s + 1.89·17-s − 1.03·19-s + 0.597·23-s + 0.200·25-s + 1.16·29-s − 0.453·31-s + 0.146·35-s − 0.0859·37-s − 1.26·41-s + 0.582·43-s − 0.202·47-s − 0.892·49-s + 1.91·53-s − 0.551·55-s + 0.894·59-s + 1.16·61-s + 0.150·65-s + 0.412·67-s + 0.381·71-s + 1.00·73-s − 0.404·77-s + 1.82·79-s + 0.707·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.077784679\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.077784679\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 - 0.867T + 7T^{2} \) |
| 11 | \( 1 + 4.08T + 11T^{2} \) |
| 13 | \( 1 - 1.21T + 13T^{2} \) |
| 17 | \( 1 - 7.82T + 17T^{2} \) |
| 19 | \( 1 + 4.51T + 19T^{2} \) |
| 23 | \( 1 - 2.86T + 23T^{2} \) |
| 29 | \( 1 - 6.29T + 29T^{2} \) |
| 31 | \( 1 + 2.52T + 31T^{2} \) |
| 37 | \( 1 + 0.523T + 37T^{2} \) |
| 41 | \( 1 + 8.12T + 41T^{2} \) |
| 43 | \( 1 - 3.82T + 43T^{2} \) |
| 47 | \( 1 + 1.39T + 47T^{2} \) |
| 53 | \( 1 - 13.9T + 53T^{2} \) |
| 59 | \( 1 - 6.87T + 59T^{2} \) |
| 61 | \( 1 - 9.08T + 61T^{2} \) |
| 67 | \( 1 - 3.37T + 67T^{2} \) |
| 71 | \( 1 - 3.21T + 71T^{2} \) |
| 73 | \( 1 - 8.60T + 73T^{2} \) |
| 79 | \( 1 - 16.2T + 79T^{2} \) |
| 83 | \( 1 - 6.44T + 83T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 + 8.77T + 97T^{2} \) |
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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
−8.385131721213605647722104278738, −8.136714968489790950006321366925, −7.16760604124014672362937337858, −6.40043546381305178489131900913, −5.37694125650044841133974329615, −5.14170540836934730154882498559, −3.90508909815520774141186752130, −2.98207464108058208539122620428, −2.08731817679573155969198663590, −0.882826897293640083917944520406,
0.882826897293640083917944520406, 2.08731817679573155969198663590, 2.98207464108058208539122620428, 3.90508909815520774141186752130, 5.14170540836934730154882498559, 5.37694125650044841133974329615, 6.40043546381305178489131900913, 7.16760604124014672362937337858, 8.136714968489790950006321366925, 8.385131721213605647722104278738
