| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s + 5·5-s + 6·6-s + 2.44·7-s + 8·8-s + 9·9-s + 10·10-s + 25.5·11-s + 12·12-s − 73.9·13-s + 4.89·14-s + 15·15-s + 16·16-s + 53.6·17-s + 18·18-s − 33.0·19-s + 20·20-s + 7.33·21-s + 51.1·22-s + 189.·23-s + 24·24-s + 25·25-s − 147.·26-s + 27·27-s + 9.78·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 0.132·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.701·11-s + 0.288·12-s − 1.57·13-s + 0.0933·14-s + 0.258·15-s + 0.250·16-s + 0.766·17-s + 0.235·18-s − 0.398·19-s + 0.223·20-s + 0.0762·21-s + 0.495·22-s + 1.71·23-s + 0.204·24-s + 0.200·25-s − 1.11·26-s + 0.192·27-s + 0.0660·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.890796553\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.890796553\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 - 3T \) |
| 5 | \( 1 - 5T \) |
| 31 | \( 1 - 31T \) |
| good | 7 | \( 1 - 2.44T + 343T^{2} \) |
| 11 | \( 1 - 25.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 73.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 53.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 33.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 189.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 247.T + 2.43e4T^{2} \) |
| 37 | \( 1 + 91.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 510.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 156.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 174.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 202.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 603.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 545.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 947.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 486.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 979.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 212.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 291.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 243.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 683.T + 9.12e5T^{2} \) |
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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
−9.662964155003185899629133757920, −8.959390092455241343347346398238, −7.84713290693159650056050242343, −7.07639720825996051063315695441, −6.26228442545647969774739879242, −5.08986512783435957623730035789, −4.45742426749114393988551300860, −3.16557386660950350071284963386, −2.41395145554959314049646512796, −1.13008083880916530277942563477,
1.13008083880916530277942563477, 2.41395145554959314049646512796, 3.16557386660950350071284963386, 4.45742426749114393988551300860, 5.08986512783435957623730035789, 6.26228442545647969774739879242, 7.07639720825996051063315695441, 7.84713290693159650056050242343, 8.959390092455241343347346398238, 9.662964155003185899629133757920
