L(s) = 1 | + 5-s + 2.82·7-s − 3·9-s + 4·11-s + 4.82·13-s + 7.65·17-s − 19-s + 2.82·23-s + 25-s − 3.65·29-s − 5.65·31-s + 2.82·35-s − 6.48·37-s − 3.65·41-s + 8.48·43-s − 3·45-s − 5.17·47-s + 1.00·49-s + 7.17·53-s + 4·55-s − 9.65·59-s + 6·61-s − 8.48·63-s + 4.82·65-s − 11.3·67-s − 5.65·71-s + 15.6·73-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 1.06·7-s − 9-s + 1.20·11-s + 1.33·13-s + 1.85·17-s − 0.229·19-s + 0.589·23-s + 0.200·25-s − 0.679·29-s − 1.01·31-s + 0.478·35-s − 1.06·37-s − 0.571·41-s + 1.29·43-s − 0.447·45-s − 0.754·47-s + 0.142·49-s + 0.985·53-s + 0.539·55-s − 1.25·59-s + 0.768·61-s − 1.06·63-s + 0.598·65-s − 1.38·67-s − 0.671·71-s + 1.83·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.630554430\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.630554430\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + 3T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 4.82T + 13T^{2} \) |
| 17 | \( 1 - 7.65T + 17T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 + 3.65T + 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 + 6.48T + 37T^{2} \) |
| 41 | \( 1 + 3.65T + 41T^{2} \) |
| 43 | \( 1 - 8.48T + 43T^{2} \) |
| 47 | \( 1 + 5.17T + 47T^{2} \) |
| 53 | \( 1 - 7.17T + 53T^{2} \) |
| 59 | \( 1 + 9.65T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 73 | \( 1 - 15.6T + 73T^{2} \) |
| 79 | \( 1 - 5.65T + 79T^{2} \) |
| 83 | \( 1 + 5.17T + 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 - 7.17T + 97T^{2} \) |
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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
−8.805233075571186055048632497434, −8.066212402028142134790658834844, −7.29644067442498535136339469534, −6.26500236092917154340128653497, −5.68430432644761039515805182789, −5.04308427901035229952216082834, −3.83645826523414314899856380878, −3.23695499460751725303895006977, −1.83626422047082080787737782402, −1.10753617760283147762944055380,
1.10753617760283147762944055380, 1.83626422047082080787737782402, 3.23695499460751725303895006977, 3.83645826523414314899856380878, 5.04308427901035229952216082834, 5.68430432644761039515805182789, 6.26500236092917154340128653497, 7.29644067442498535136339469534, 8.066212402028142134790658834844, 8.805233075571186055048632497434