| L(s) = 1 | + 1.65·2-s + 3-s + 0.728·4-s − 5-s + 1.65·6-s + 3.71·7-s − 2.09·8-s + 9-s − 1.65·10-s + 6.46·11-s + 0.728·12-s + 6.46·13-s + 6.14·14-s − 15-s − 4.92·16-s + 5.31·17-s + 1.65·18-s + 0.400·19-s − 0.728·20-s + 3.71·21-s + 10.6·22-s − 7.98·23-s − 2.09·24-s + 25-s + 10.6·26-s + 27-s + 2.71·28-s + ⋯ |
| L(s) = 1 | + 1.16·2-s + 0.577·3-s + 0.364·4-s − 0.447·5-s + 0.674·6-s + 1.40·7-s − 0.742·8-s + 0.333·9-s − 0.522·10-s + 1.94·11-s + 0.210·12-s + 1.79·13-s + 1.64·14-s − 0.258·15-s − 1.23·16-s + 1.28·17-s + 0.389·18-s + 0.0917·19-s − 0.162·20-s + 0.811·21-s + 2.27·22-s − 1.66·23-s − 0.428·24-s + 0.200·25-s + 2.09·26-s + 0.192·27-s + 0.512·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.784999161\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.784999161\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 - 1.65T + 2T^{2} \) |
| 7 | \( 1 - 3.71T + 7T^{2} \) |
| 11 | \( 1 - 6.46T + 11T^{2} \) |
| 13 | \( 1 - 6.46T + 13T^{2} \) |
| 17 | \( 1 - 5.31T + 17T^{2} \) |
| 19 | \( 1 - 0.400T + 19T^{2} \) |
| 23 | \( 1 + 7.98T + 23T^{2} \) |
| 29 | \( 1 + 6.37T + 29T^{2} \) |
| 31 | \( 1 - 3.38T + 31T^{2} \) |
| 37 | \( 1 - 1.38T + 37T^{2} \) |
| 41 | \( 1 + 4.50T + 41T^{2} \) |
| 43 | \( 1 - 2.08T + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 - 2.51T + 53T^{2} \) |
| 59 | \( 1 + 4.18T + 59T^{2} \) |
| 61 | \( 1 - 7.53T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 + 4.61T + 71T^{2} \) |
| 73 | \( 1 + 9.21T + 73T^{2} \) |
| 79 | \( 1 + 8.21T + 79T^{2} \) |
| 83 | \( 1 - 3.62T + 83T^{2} \) |
| 89 | \( 1 + 3.58T + 89T^{2} \) |
| 97 | \( 1 + 0.241T + 97T^{2} \) |
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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
−8.302169902647365816905541121579, −7.36698687569838608604353451346, −6.39039287536296765507964636521, −5.91738023854892694996372929047, −5.05415204025590461936169546733, −4.11872979398125687258105247186, −3.87302511772069202123751898910, −3.27278069540732574866873924840, −1.83707635019816023107802955326, −1.19525971966412443899652762761,
1.19525971966412443899652762761, 1.83707635019816023107802955326, 3.27278069540732574866873924840, 3.87302511772069202123751898910, 4.11872979398125687258105247186, 5.05415204025590461936169546733, 5.91738023854892694996372929047, 6.39039287536296765507964636521, 7.36698687569838608604353451346, 8.302169902647365816905541121579
