| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s + 13.7·5-s − 6·6-s + 7·7-s + 8·8-s + 9·9-s + 27.4·10-s + 45.9·11-s − 12·12-s + 5.45·13-s + 14·14-s − 41.1·15-s + 16·16-s + 39.2·17-s + 18·18-s + 27.3·19-s + 54.9·20-s − 21·21-s + 91.8·22-s − 23·23-s − 24·24-s + 63.5·25-s + 10.9·26-s − 27·27-s + 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.22·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.868·10-s + 1.25·11-s − 0.288·12-s + 0.116·13-s + 0.267·14-s − 0.709·15-s + 0.250·16-s + 0.560·17-s + 0.235·18-s + 0.330·19-s + 0.614·20-s − 0.218·21-s + 0.889·22-s − 0.208·23-s − 0.204·24-s + 0.508·25-s + 0.0822·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.247061004\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.247061004\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 3T \) |
| 7 | \( 1 - 7T \) |
| 23 | \( 1 + 23T \) |
| good | 5 | \( 1 - 13.7T + 125T^{2} \) |
| 11 | \( 1 - 45.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 5.45T + 2.19e3T^{2} \) |
| 17 | \( 1 - 39.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 27.3T + 6.85e3T^{2} \) |
| 29 | \( 1 - 9.12T + 2.43e4T^{2} \) |
| 31 | \( 1 - 29.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 350.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 445.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 447.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 585.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 595.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 365.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 629.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 150.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 125.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 702.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 783.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 744.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 40.6T + 7.04e5T^{2} \) |
| 97 | \( 1 - 394.T + 9.12e5T^{2} \) |
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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
−9.808301389746525304474184551762, −8.979276058752895934320223806555, −7.77992650524409667239707981206, −6.67941682573029616787829204164, −6.12606743677945289657751299493, −5.36394958771885761063335484562, −4.49724108013594244841814373051, −3.38001188915950982271562424579, −1.99546644446973938367685688624, −1.14099143173492481957862385567,
1.14099143173492481957862385567, 1.99546644446973938367685688624, 3.38001188915950982271562424579, 4.49724108013594244841814373051, 5.36394958771885761063335484562, 6.12606743677945289657751299493, 6.67941682573029616787829204164, 7.77992650524409667239707981206, 8.979276058752895934320223806555, 9.808301389746525304474184551762
