| L(s) = 1 | − 2-s + 4-s − 1.51·5-s + 7-s − 8-s + 1.51·10-s − 3.51·11-s + 6.28·13-s − 14-s + 16-s + 6.28·17-s + 4.76·19-s − 1.51·20-s + 3.51·22-s − 23-s − 2.69·25-s − 6.28·26-s + 28-s − 2·29-s + 1.41·31-s − 32-s − 6.28·34-s − 1.51·35-s − 6.39·37-s − 4.76·38-s + 1.51·40-s + 2·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.678·5-s + 0.377·7-s − 0.353·8-s + 0.479·10-s − 1.06·11-s + 1.74·13-s − 0.267·14-s + 0.250·16-s + 1.52·17-s + 1.09·19-s − 0.339·20-s + 0.749·22-s − 0.208·23-s − 0.539·25-s − 1.23·26-s + 0.188·28-s − 0.371·29-s + 0.253·31-s − 0.176·32-s − 1.07·34-s − 0.256·35-s − 1.05·37-s − 0.773·38-s + 0.239·40-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.218294404\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.218294404\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 1.51T + 5T^{2} \) |
| 11 | \( 1 + 3.51T + 11T^{2} \) |
| 13 | \( 1 - 6.28T + 13T^{2} \) |
| 17 | \( 1 - 6.28T + 17T^{2} \) |
| 19 | \( 1 - 4.76T + 19T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 1.41T + 31T^{2} \) |
| 37 | \( 1 + 6.39T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 12.0T + 43T^{2} \) |
| 47 | \( 1 - 2.58T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 - 7.03T + 59T^{2} \) |
| 61 | \( 1 + 4.55T + 61T^{2} \) |
| 67 | \( 1 - 14.0T + 67T^{2} \) |
| 71 | \( 1 - 9.90T + 71T^{2} \) |
| 73 | \( 1 - 8.87T + 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 - 3.59T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 + 7.46T + 97T^{2} \) |
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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
−8.501261571274504636482262829407, −8.035428553381949756330794030914, −7.62064783365414449134385467511, −6.63886470458506352847278460361, −5.68081890083384192522813221255, −5.08595223063919992837121817678, −3.68127643495281512115853157191, −3.25400610072391066015209302760, −1.82573917371782213814664469228, −0.77869790658016497518051515454,
0.77869790658016497518051515454, 1.82573917371782213814664469228, 3.25400610072391066015209302760, 3.68127643495281512115853157191, 5.08595223063919992837121817678, 5.68081890083384192522813221255, 6.63886470458506352847278460361, 7.62064783365414449134385467511, 8.035428553381949756330794030914, 8.501261571274504636482262829407
