| L(s) = 1 | − 2·2-s − 1.34·3-s + 4·4-s + 5·5-s + 2.69·6-s + 32.8·7-s − 8·8-s − 25.1·9-s − 10·10-s + 58.9·11-s − 5.38·12-s − 60.5·13-s − 65.7·14-s − 6.73·15-s + 16·16-s + 31.1·17-s + 50.3·18-s − 43.2·19-s + 20·20-s − 44.2·21-s − 117.·22-s + 23·23-s + 10.7·24-s + 25·25-s + 121.·26-s + 70.3·27-s + 131.·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.259·3-s + 0.5·4-s + 0.447·5-s + 0.183·6-s + 1.77·7-s − 0.353·8-s − 0.932·9-s − 0.316·10-s + 1.61·11-s − 0.129·12-s − 1.29·13-s − 1.25·14-s − 0.115·15-s + 0.250·16-s + 0.444·17-s + 0.659·18-s − 0.522·19-s + 0.223·20-s − 0.460·21-s − 1.14·22-s + 0.208·23-s + 0.0916·24-s + 0.200·25-s + 0.913·26-s + 0.501·27-s + 0.887·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.511445923\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.511445923\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 5 | \( 1 - 5T \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 + 1.34T + 27T^{2} \) |
| 7 | \( 1 - 32.8T + 343T^{2} \) |
| 11 | \( 1 - 58.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 60.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 31.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 43.2T + 6.85e3T^{2} \) |
| 29 | \( 1 + 13.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 287.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 279.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 419.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 421.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 117.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 439.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 248.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 186.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 888.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 444.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 925.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 755.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 648.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 542.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 685.T + 9.12e5T^{2} \) |
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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
−11.68687568606478360227231684829, −10.87778504000389198839920528437, −9.754437602719335880740009251598, −8.768205236052827149289716323440, −7.984413571632123416122587518040, −6.79333650796890280496185443296, −5.61057214595317156839837665673, −4.47427118315126127356962711490, −2.41525348158693546969786872192, −1.10780033761726753687201787193,
1.10780033761726753687201787193, 2.41525348158693546969786872192, 4.47427118315126127356962711490, 5.61057214595317156839837665673, 6.79333650796890280496185443296, 7.984413571632123416122587518040, 8.768205236052827149289716323440, 9.754437602719335880740009251598, 10.87778504000389198839920528437, 11.68687568606478360227231684829
