| L(s) = 1 | − 2.93·3-s − 5-s + 1.31·7-s + 5.63·9-s + 0.258·11-s − 5.87·13-s + 2.93·15-s + 4.25·17-s − 2.93·19-s − 3.87·21-s + 8.51·23-s + 25-s − 7.75·27-s − 3.61·29-s − 3.95·31-s − 0.760·33-s − 1.31·35-s − 37-s + 17.2·39-s − 8.89·41-s + 5.61·43-s − 5.63·45-s − 5.57·47-s − 5.25·49-s − 12.5·51-s + 12.8·53-s − 0.258·55-s + ⋯ |
| L(s) = 1 | − 1.69·3-s − 0.447·5-s + 0.498·7-s + 1.87·9-s + 0.0780·11-s − 1.63·13-s + 0.758·15-s + 1.03·17-s − 0.674·19-s − 0.846·21-s + 1.77·23-s + 0.200·25-s − 1.49·27-s − 0.672·29-s − 0.710·31-s − 0.132·33-s − 0.223·35-s − 0.164·37-s + 2.76·39-s − 1.38·41-s + 0.856·43-s − 0.840·45-s − 0.813·47-s − 0.751·49-s − 1.75·51-s + 1.77·53-s − 0.0348·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6833580474\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6833580474\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 + 2.93T + 3T^{2} \) |
| 7 | \( 1 - 1.31T + 7T^{2} \) |
| 11 | \( 1 - 0.258T + 11T^{2} \) |
| 13 | \( 1 + 5.87T + 13T^{2} \) |
| 17 | \( 1 - 4.25T + 17T^{2} \) |
| 19 | \( 1 + 2.93T + 19T^{2} \) |
| 23 | \( 1 - 8.51T + 23T^{2} \) |
| 29 | \( 1 + 3.61T + 29T^{2} \) |
| 31 | \( 1 + 3.95T + 31T^{2} \) |
| 41 | \( 1 + 8.89T + 41T^{2} \) |
| 43 | \( 1 - 5.61T + 43T^{2} \) |
| 47 | \( 1 + 5.57T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 + 9.57T + 59T^{2} \) |
| 61 | \( 1 - 0.380T + 61T^{2} \) |
| 67 | \( 1 + 5.57T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 - 2.51T + 73T^{2} \) |
| 79 | \( 1 - 7.69T + 79T^{2} \) |
| 83 | \( 1 - 6.17T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−8.789374647862037600867337495180, −7.61552873696836340248691502387, −7.24453615851395603015126479886, −6.46801628707090951076647729235, −5.47004372377051382802112506760, −5.02602647991771363589310278098, −4.40148377243558386877260217612, −3.19691709010331543783900725482, −1.76792687656029780591772315617, −0.55472076720297396479597959243,
0.55472076720297396479597959243, 1.76792687656029780591772315617, 3.19691709010331543783900725482, 4.40148377243558386877260217612, 5.02602647991771363589310278098, 5.47004372377051382802112506760, 6.46801628707090951076647729235, 7.24453615851395603015126479886, 7.61552873696836340248691502387, 8.789374647862037600867337495180
