| L(s) = 1 | − 2.93·2-s − 3.85·3-s + 0.637·4-s − 5·5-s + 11.3·6-s − 23.5·7-s + 21.6·8-s − 12.1·9-s + 14.6·10-s − 58.2·11-s − 2.45·12-s + 68.5·13-s + 69.3·14-s + 19.2·15-s − 68.6·16-s + 101.·17-s + 35.6·18-s − 7.02·19-s − 3.18·20-s + 91.0·21-s + 171.·22-s − 23·23-s − 83.4·24-s + 25·25-s − 201.·26-s + 150.·27-s − 15.0·28-s + ⋯ |
| L(s) = 1 | − 1.03·2-s − 0.742·3-s + 0.0797·4-s − 0.447·5-s + 0.771·6-s − 1.27·7-s + 0.956·8-s − 0.448·9-s + 0.464·10-s − 1.59·11-s − 0.0591·12-s + 1.46·13-s + 1.32·14-s + 0.332·15-s − 1.07·16-s + 1.44·17-s + 0.466·18-s − 0.0848·19-s − 0.0356·20-s + 0.945·21-s + 1.65·22-s − 0.208·23-s − 0.709·24-s + 0.200·25-s − 1.51·26-s + 1.07·27-s − 0.101·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3393370062\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3393370062\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 23 | \( 1 + 23T \) |
| good | 2 | \( 1 + 2.93T + 8T^{2} \) |
| 3 | \( 1 + 3.85T + 27T^{2} \) |
| 7 | \( 1 + 23.5T + 343T^{2} \) |
| 11 | \( 1 + 58.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 68.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 101.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 7.02T + 6.85e3T^{2} \) |
| 29 | \( 1 - 206.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 54.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 241.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 122.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 320.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 107.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 127.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 693.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 899.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 110.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 225.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 746.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.09e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.28e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.20e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 903.T + 9.12e5T^{2} \) |
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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
−12.99298136631232983892396359156, −11.88669253397764018252067523996, −10.57890989989705518926454502166, −10.16209450217894625077240365764, −8.715476569558740201865894845728, −7.890627998094180743156842988206, −6.45859579729192296268086441367, −5.22482506741171296884994348225, −3.29939336007389935685831012019, −0.59393817036756763483328392026,
0.59393817036756763483328392026, 3.29939336007389935685831012019, 5.22482506741171296884994348225, 6.45859579729192296268086441367, 7.890627998094180743156842988206, 8.715476569558740201865894845728, 10.16209450217894625077240365764, 10.57890989989705518926454502166, 11.88669253397764018252067523996, 12.99298136631232983892396359156
