| L(s) = 1 | + 2·2-s + 9.32·3-s + 4·4-s + 18.6·6-s + 14.4·7-s + 8·8-s + 59.9·9-s − 51.5·11-s + 37.2·12-s + 2.32·13-s + 28.9·14-s + 16·16-s + 88.7·17-s + 119.·18-s + 7.15·19-s + 134.·21-s − 103.·22-s − 23·23-s + 74.5·24-s + 4.65·26-s + 307.·27-s + 57.8·28-s + 88.6·29-s − 79.1·31-s + 32·32-s − 480.·33-s + 177.·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.79·3-s + 0.5·4-s + 1.26·6-s + 0.780·7-s + 0.353·8-s + 2.21·9-s − 1.41·11-s + 0.897·12-s + 0.0496·13-s + 0.551·14-s + 0.250·16-s + 1.26·17-s + 1.56·18-s + 0.0863·19-s + 1.40·21-s − 0.999·22-s − 0.208·23-s + 0.634·24-s + 0.0351·26-s + 2.18·27-s + 0.390·28-s + 0.567·29-s − 0.458·31-s + 0.176·32-s − 2.53·33-s + 0.895·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.420905099\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.420905099\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 - 9.32T + 27T^{2} \) |
| 7 | \( 1 - 14.4T + 343T^{2} \) |
| 11 | \( 1 + 51.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 2.32T + 2.19e3T^{2} \) |
| 17 | \( 1 - 88.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 7.15T + 6.85e3T^{2} \) |
| 29 | \( 1 - 88.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 79.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 435.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 58.6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 145.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 105.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 394.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 333.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 713.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 536.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 678.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 618.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.18e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.23e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 811.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 833.T + 9.12e5T^{2} \) |
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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
−9.371118507261791942473386678066, −8.340751490390587364051692188404, −7.78694869848589247712714313592, −7.37267072612606451513509272937, −5.90525690809791864427450664237, −4.90270669640398610932043917105, −4.06548825016134683736432899623, −3.02374456975728765206226663275, −2.45167686252437009889784949064, −1.32566811156042646259745407384,
1.32566811156042646259745407384, 2.45167686252437009889784949064, 3.02374456975728765206226663275, 4.06548825016134683736432899623, 4.90270669640398610932043917105, 5.90525690809791864427450664237, 7.37267072612606451513509272937, 7.78694869848589247712714313592, 8.340751490390587364051692188404, 9.371118507261791942473386678066
