| L(s) = 1 | + 2.27·2-s − 0.518·3-s + 3.15·4-s + 5-s − 1.17·6-s − 1.33·7-s + 2.62·8-s − 2.73·9-s + 2.27·10-s + 5.55·11-s − 1.63·12-s − 2.96·13-s − 3.03·14-s − 0.518·15-s − 0.346·16-s − 0.426·17-s − 6.20·18-s − 5.57·19-s + 3.15·20-s + 0.692·21-s + 12.6·22-s − 3.50·23-s − 1.36·24-s + 25-s − 6.74·26-s + 2.97·27-s − 4.21·28-s + ⋯ |
| L(s) = 1 | + 1.60·2-s − 0.299·3-s + 1.57·4-s + 0.447·5-s − 0.481·6-s − 0.504·7-s + 0.929·8-s − 0.910·9-s + 0.718·10-s + 1.67·11-s − 0.472·12-s − 0.823·13-s − 0.810·14-s − 0.133·15-s − 0.0867·16-s − 0.103·17-s − 1.46·18-s − 1.27·19-s + 0.705·20-s + 0.151·21-s + 2.68·22-s − 0.730·23-s − 0.278·24-s + 0.200·25-s − 1.32·26-s + 0.572·27-s − 0.796·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.338323945\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.338323945\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 - 2.27T + 2T^{2} \) |
| 3 | \( 1 + 0.518T + 3T^{2} \) |
| 7 | \( 1 + 1.33T + 7T^{2} \) |
| 11 | \( 1 - 5.55T + 11T^{2} \) |
| 13 | \( 1 + 2.96T + 13T^{2} \) |
| 17 | \( 1 + 0.426T + 17T^{2} \) |
| 19 | \( 1 + 5.57T + 19T^{2} \) |
| 23 | \( 1 + 3.50T + 23T^{2} \) |
| 29 | \( 1 - 8.06T + 29T^{2} \) |
| 31 | \( 1 - 4.57T + 31T^{2} \) |
| 41 | \( 1 + 6.87T + 41T^{2} \) |
| 43 | \( 1 - 7.35T + 43T^{2} \) |
| 47 | \( 1 - 3.16T + 47T^{2} \) |
| 53 | \( 1 + 6.51T + 53T^{2} \) |
| 59 | \( 1 - 8.51T + 59T^{2} \) |
| 61 | \( 1 - 3.31T + 61T^{2} \) |
| 67 | \( 1 - 9.68T + 67T^{2} \) |
| 71 | \( 1 - 7.81T + 71T^{2} \) |
| 73 | \( 1 + 0.762T + 73T^{2} \) |
| 79 | \( 1 - 17.0T + 79T^{2} \) |
| 83 | \( 1 + 2.74T + 83T^{2} \) |
| 89 | \( 1 - 0.0865T + 89T^{2} \) |
| 97 | \( 1 + 17.8T + 97T^{2} \) |
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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
−12.47729762951628012534597075436, −12.05631209440979613921304880113, −11.08307015190951594008641239630, −9.783296993118674620572568407089, −8.588080735074930026816576287641, −6.58574219437354697796779912855, −6.30450994648643114565642133435, −5.00801190681046763553393171217, −3.88329977642399612327704776499, −2.50000991822749863152064229050,
2.50000991822749863152064229050, 3.88329977642399612327704776499, 5.00801190681046763553393171217, 6.30450994648643114565642133435, 6.58574219437354697796779912855, 8.588080735074930026816576287641, 9.783296993118674620572568407089, 11.08307015190951594008641239630, 12.05631209440979613921304880113, 12.47729762951628012534597075436
