| L(s) = 1 | + 0.0173·5-s + (−1.99 + 1.74i)7-s + 4.77i·11-s + 1.51i·13-s − 1.91·17-s + 6.45i·19-s − 4.41i·23-s − 4.99·25-s + 2.00i·29-s − 7.52i·31-s + (−0.0346 + 0.0303i)35-s − 6.10·37-s − 9.35·41-s + 2.93·43-s + 2.91·47-s + ⋯ |
| L(s) = 1 | + 0.00778·5-s + (−0.752 + 0.658i)7-s + 1.43i·11-s + 0.421i·13-s − 0.463·17-s + 1.48i·19-s − 0.920i·23-s − 0.999·25-s + 0.372i·29-s − 1.35i·31-s + (−0.00585 + 0.00512i)35-s − 1.00·37-s − 1.46·41-s + 0.447·43-s + 0.424·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.752 + 0.658i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.752 + 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1433212970\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1433212970\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.99 - 1.74i)T \) |
| good | 5 | \( 1 - 0.0173T + 5T^{2} \) |
| 11 | \( 1 - 4.77iT - 11T^{2} \) |
| 13 | \( 1 - 1.51iT - 13T^{2} \) |
| 17 | \( 1 + 1.91T + 17T^{2} \) |
| 19 | \( 1 - 6.45iT - 19T^{2} \) |
| 23 | \( 1 + 4.41iT - 23T^{2} \) |
| 29 | \( 1 - 2.00iT - 29T^{2} \) |
| 31 | \( 1 + 7.52iT - 31T^{2} \) |
| 37 | \( 1 + 6.10T + 37T^{2} \) |
| 41 | \( 1 + 9.35T + 41T^{2} \) |
| 43 | \( 1 - 2.93T + 43T^{2} \) |
| 47 | \( 1 - 2.91T + 47T^{2} \) |
| 53 | \( 1 - 8.94iT - 53T^{2} \) |
| 59 | \( 1 - 8.16T + 59T^{2} \) |
| 61 | \( 1 + 0.559iT - 61T^{2} \) |
| 67 | \( 1 - 7.38T + 67T^{2} \) |
| 71 | \( 1 + 11.6iT - 71T^{2} \) |
| 73 | \( 1 + 12.8iT - 73T^{2} \) |
| 79 | \( 1 + 9.51T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 - 4.00T + 89T^{2} \) |
| 97 | \( 1 - 4.72iT - 97T^{2} \) |
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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
−8.866863488869412154317105755258, −8.025663093303676039252288186614, −7.30682742908614256486277556997, −6.54080267207770892105518014446, −5.97622751134556198069232097126, −5.09855093217750829531160725060, −4.23434001652610974695530768760, −3.54174573783518127819096042282, −2.34290839008493062368673530555, −1.79825162993381768398742531685,
0.04247102093897975816471911526, 1.04456023466559390006339087606, 2.46681978629269220059099372551, 3.38946455364548932022239215663, 3.84016419422986228518039716508, 5.05471475603842521947377126813, 5.63777231066395820963067862450, 6.61752682092652465946461851060, 6.97439602584981405961323270021, 7.929637821898389905337751917852
