| L(s) = 1 | − 5·5-s + 23.9·7-s + 57.9·11-s − 8.16·13-s − 50.0·17-s + 69.7·19-s + 4.92·23-s + 25·25-s + 79.4·29-s + 260.·31-s − 119.·35-s − 223.·37-s − 337.·41-s + 326.·43-s + 89.6·47-s + 229.·49-s − 543.·53-s − 289.·55-s + 92·59-s + 159.·61-s + 40.8·65-s − 910.·67-s − 293.·71-s + 142.·73-s + 1.38e3·77-s + 1.10e3·79-s + 813.·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.29·7-s + 1.58·11-s − 0.174·13-s − 0.714·17-s + 0.842·19-s + 0.0446·23-s + 0.200·25-s + 0.508·29-s + 1.50·31-s − 0.577·35-s − 0.994·37-s − 1.28·41-s + 1.15·43-s + 0.278·47-s + 0.668·49-s − 1.40·53-s − 0.710·55-s + 0.203·59-s + 0.334·61-s + 0.0778·65-s − 1.66·67-s − 0.490·71-s + 0.227·73-s + 2.05·77-s + 1.57·79-s + 1.07·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.606322873\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.606322873\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 5T \) |
| good | 7 | \( 1 - 23.9T + 343T^{2} \) |
| 11 | \( 1 - 57.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 8.16T + 2.19e3T^{2} \) |
| 17 | \( 1 + 50.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 69.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 4.92T + 1.21e4T^{2} \) |
| 29 | \( 1 - 79.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 260.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 223.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 337.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 326.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 89.6T + 1.03e5T^{2} \) |
| 53 | \( 1 + 543.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 92T + 2.05e5T^{2} \) |
| 61 | \( 1 - 159.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 910.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 293.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 142.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.10e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 813.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 956.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 106.T + 9.12e5T^{2} \) |
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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
−9.339655583274460946768612330445, −8.653753170908977610822391678171, −7.907583608186401647455822067093, −7.02805181536773130156284914985, −6.22154693509243648189103404191, −4.94382816920704298364905086955, −4.37609033524121752503905947276, −3.30577056430641318174548714212, −1.87588585337426722698519022384, −0.906607480859324615864999803966,
0.906607480859324615864999803966, 1.87588585337426722698519022384, 3.30577056430641318174548714212, 4.37609033524121752503905947276, 4.94382816920704298364905086955, 6.22154693509243648189103404191, 7.02805181536773130156284914985, 7.907583608186401647455822067093, 8.653753170908977610822391678171, 9.339655583274460946768612330445
