| L(s) = 1 | + 2.18·2-s + 2.79·4-s + 3.79·7-s + 1.73·8-s − 0.456·11-s + 5.79·13-s + 8.29·14-s − 1.79·16-s − 7.48·17-s + 5.58·19-s − 0.999·22-s − 0.361·23-s + 12.6·26-s + 10.5·28-s + 5.29·29-s + 3·31-s − 7.38·32-s − 16.3·34-s + 4.79·37-s + 12.2·38-s − 6.47·41-s − 3.37·43-s − 1.27·44-s − 0.791·46-s − 6.10·47-s + 7.37·49-s + 16.1·52-s + ⋯ |
| L(s) = 1 | + 1.54·2-s + 1.39·4-s + 1.43·7-s + 0.612·8-s − 0.137·11-s + 1.60·13-s + 2.21·14-s − 0.447·16-s − 1.81·17-s + 1.28·19-s − 0.213·22-s − 0.0753·23-s + 2.48·26-s + 1.99·28-s + 0.982·29-s + 0.538·31-s − 1.30·32-s − 2.80·34-s + 0.787·37-s + 1.98·38-s − 1.01·41-s − 0.514·43-s − 0.192·44-s − 0.116·46-s − 0.891·47-s + 1.05·49-s + 2.24·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.007869434\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.007869434\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 2.18T + 2T^{2} \) |
| 7 | \( 1 - 3.79T + 7T^{2} \) |
| 11 | \( 1 + 0.456T + 11T^{2} \) |
| 13 | \( 1 - 5.79T + 13T^{2} \) |
| 17 | \( 1 + 7.48T + 17T^{2} \) |
| 19 | \( 1 - 5.58T + 19T^{2} \) |
| 23 | \( 1 + 0.361T + 23T^{2} \) |
| 29 | \( 1 - 5.29T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 - 4.79T + 37T^{2} \) |
| 41 | \( 1 + 6.47T + 41T^{2} \) |
| 43 | \( 1 + 3.37T + 43T^{2} \) |
| 47 | \( 1 + 6.10T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 - 8.20T + 59T^{2} \) |
| 61 | \( 1 + 7.37T + 61T^{2} \) |
| 67 | \( 1 - 1.41T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 + 4.16T + 73T^{2} \) |
| 79 | \( 1 + 15.1T + 79T^{2} \) |
| 83 | \( 1 - 5.19T + 83T^{2} \) |
| 89 | \( 1 - 2.74T + 89T^{2} \) |
| 97 | \( 1 + 5.16T + 97T^{2} \) |
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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
−8.796500028712644884689720033483, −8.455144063456203355379333122801, −7.35266804298045625104413158595, −6.50313163483949715489408340821, −5.79751600310262488632957488625, −4.90346258810510879206454211901, −4.43167461754631413956279969384, −3.52860954545473017983365908833, −2.48369088098561818909828567514, −1.38738661403581394858376937450,
1.38738661403581394858376937450, 2.48369088098561818909828567514, 3.52860954545473017983365908833, 4.43167461754631413956279969384, 4.90346258810510879206454211901, 5.79751600310262488632957488625, 6.50313163483949715489408340821, 7.35266804298045625104413158595, 8.455144063456203355379333122801, 8.796500028712644884689720033483
