| L(s) = 1 | − 1.33·2-s − 6.23·4-s + 10.6·7-s + 18.9·8-s − 11.2·11-s − 2.74·13-s − 14.2·14-s + 24.6·16-s − 29.5·17-s − 31.1·19-s + 14.9·22-s + 116.·23-s + 3.64·26-s − 66.6·28-s − 108.·29-s + 70.7·31-s − 184.·32-s + 39.3·34-s − 282.·37-s + 41.3·38-s + 425.·41-s − 312.·43-s + 70.1·44-s − 155.·46-s − 193.·47-s − 228.·49-s + 17.0·52-s + ⋯ |
| L(s) = 1 | − 0.470·2-s − 0.778·4-s + 0.577·7-s + 0.836·8-s − 0.308·11-s − 0.0584·13-s − 0.271·14-s + 0.385·16-s − 0.421·17-s − 0.375·19-s + 0.145·22-s + 1.06·23-s + 0.0274·26-s − 0.449·28-s − 0.694·29-s + 0.410·31-s − 1.01·32-s + 0.198·34-s − 1.25·37-s + 0.176·38-s + 1.62·41-s − 1.10·43-s + 0.240·44-s − 0.498·46-s − 0.600·47-s − 0.666·49-s + 0.0455·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.153220613\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.153220613\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 1.33T + 8T^{2} \) |
| 7 | \( 1 - 10.6T + 343T^{2} \) |
| 11 | \( 1 + 11.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 2.74T + 2.19e3T^{2} \) |
| 17 | \( 1 + 29.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 31.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 116.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 108.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 70.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 282.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 425.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 312.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 193.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 103.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 494.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 424.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 586.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.13e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 302.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 525.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.00e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.42e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 25.6T + 9.12e5T^{2} \) |
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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
−10.02824776389431903476765152638, −9.147513842520129405462946499821, −8.456748879818007086605541141057, −7.71093757724792710960872939975, −6.71464598634358418014521143382, −5.34669989185591881743406908936, −4.69152601902512681012189090573, −3.57446757180698806678877815931, −2.02067237944850840104989279390, −0.68111586178742939065748965356,
0.68111586178742939065748965356, 2.02067237944850840104989279390, 3.57446757180698806678877815931, 4.69152601902512681012189090573, 5.34669989185591881743406908936, 6.71464598634358418014521143382, 7.71093757724792710960872939975, 8.456748879818007086605541141057, 9.147513842520129405462946499821, 10.02824776389431903476765152638
