| L(s) = 1 | + 4.21·3-s + 5·5-s + 19.9·7-s − 9.20·9-s − 0.532·11-s − 41.4·13-s + 21.0·15-s + 60.4·17-s − 69.0·19-s + 83.9·21-s − 23·23-s + 25·25-s − 152.·27-s + 109.·29-s + 128.·31-s − 2.24·33-s + 99.5·35-s + 139.·37-s − 174.·39-s + 488.·41-s + 197.·43-s − 46.0·45-s − 447.·47-s + 53.3·49-s + 254.·51-s + 420.·53-s − 2.66·55-s + ⋯ |
| L(s) = 1 | + 0.811·3-s + 0.447·5-s + 1.07·7-s − 0.340·9-s − 0.0145·11-s − 0.883·13-s + 0.363·15-s + 0.861·17-s − 0.834·19-s + 0.872·21-s − 0.208·23-s + 0.200·25-s − 1.08·27-s + 0.703·29-s + 0.744·31-s − 0.0118·33-s + 0.480·35-s + 0.619·37-s − 0.717·39-s + 1.86·41-s + 0.700·43-s − 0.152·45-s − 1.38·47-s + 0.155·49-s + 0.699·51-s + 1.09·53-s − 0.00652·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.596370819\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.596370819\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 - 4.21T + 27T^{2} \) |
| 7 | \( 1 - 19.9T + 343T^{2} \) |
| 11 | \( 1 + 0.532T + 1.33e3T^{2} \) |
| 13 | \( 1 + 41.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 60.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 69.0T + 6.85e3T^{2} \) |
| 29 | \( 1 - 109.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 128.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 139.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 488.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 197.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 447.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 420.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 728.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 528.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 300.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 593.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 996.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 3.39T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.03e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 261.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.65e3T + 9.12e5T^{2} \) |
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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
−8.785065618401374466027318634290, −8.091377875926839708842874585849, −7.64442874969564633565220053280, −6.52222308143676618328130985625, −5.58120539636607569501448738229, −4.81676356862520841792686076212, −3.88007775258642385120912708847, −2.65850305039670375424294482375, −2.13714247094967952376546626788, −0.860763009186747042563098181198,
0.860763009186747042563098181198, 2.13714247094967952376546626788, 2.65850305039670375424294482375, 3.88007775258642385120912708847, 4.81676356862520841792686076212, 5.58120539636607569501448738229, 6.52222308143676618328130985625, 7.64442874969564633565220053280, 8.091377875926839708842874585849, 8.785065618401374466027318634290
