| L(s) = 1 | + 2.60·3-s + 1.72·5-s − 7-s + 3.78·9-s + 5.39·11-s − 6.59·13-s + 4.50·15-s + 0.787·17-s + 1.71·19-s − 2.60·21-s − 4.31·23-s − 2.01·25-s + 2.05·27-s + 8.40·29-s + 8.32·31-s + 14.0·33-s − 1.72·35-s − 3.61·37-s − 17.1·39-s − 9.74·41-s + 11.6·43-s + 6.54·45-s + 2.84·47-s + 49-s + 2.05·51-s + 6.70·53-s + 9.32·55-s + ⋯ |
| L(s) = 1 | + 1.50·3-s + 0.772·5-s − 0.377·7-s + 1.26·9-s + 1.62·11-s − 1.82·13-s + 1.16·15-s + 0.190·17-s + 0.394·19-s − 0.568·21-s − 0.899·23-s − 0.403·25-s + 0.394·27-s + 1.56·29-s + 1.49·31-s + 2.44·33-s − 0.291·35-s − 0.594·37-s − 2.75·39-s − 1.52·41-s + 1.77·43-s + 0.975·45-s + 0.415·47-s + 0.142·49-s + 0.287·51-s + 0.921·53-s + 1.25·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.543467560\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.543467560\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 89 | \( 1 + T \) |
| good | 3 | \( 1 - 2.60T + 3T^{2} \) |
| 5 | \( 1 - 1.72T + 5T^{2} \) |
| 11 | \( 1 - 5.39T + 11T^{2} \) |
| 13 | \( 1 + 6.59T + 13T^{2} \) |
| 17 | \( 1 - 0.787T + 17T^{2} \) |
| 19 | \( 1 - 1.71T + 19T^{2} \) |
| 23 | \( 1 + 4.31T + 23T^{2} \) |
| 29 | \( 1 - 8.40T + 29T^{2} \) |
| 31 | \( 1 - 8.32T + 31T^{2} \) |
| 37 | \( 1 + 3.61T + 37T^{2} \) |
| 41 | \( 1 + 9.74T + 41T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 - 2.84T + 47T^{2} \) |
| 53 | \( 1 - 6.70T + 53T^{2} \) |
| 59 | \( 1 - 2.34T + 59T^{2} \) |
| 61 | \( 1 - 2.07T + 61T^{2} \) |
| 67 | \( 1 - 7.59T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 - 9.16T + 73T^{2} \) |
| 79 | \( 1 + 14.9T + 79T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 97 | \( 1 - 9.50T + 97T^{2} \) |
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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
−7.77639499865251009413156542608, −6.94241648478038608192767093801, −6.55385045938150299100214295657, −5.65566370421191778455627609174, −4.73614773566089114261416774434, −4.02113139107037078621484109185, −3.32041097046816811202370013374, −2.46127267328964588726196509377, −2.07758983874170239341874324112, −0.952856820204657380116712525039,
0.952856820204657380116712525039, 2.07758983874170239341874324112, 2.46127267328964588726196509377, 3.32041097046816811202370013374, 4.02113139107037078621484109185, 4.73614773566089114261416774434, 5.65566370421191778455627609174, 6.55385045938150299100214295657, 6.94241648478038608192767093801, 7.77639499865251009413156542608
