| L(s) = 1 | + 3.05·3-s − 3.77·5-s − 1.23·7-s + 6.31·9-s − 6.24·11-s + 1.40·13-s − 11.5·15-s + 17-s − 5.85·19-s − 3.77·21-s − 4.79·23-s + 9.25·25-s + 10.1·27-s + 3.61·29-s + 7.20·31-s − 19.0·33-s + 4.67·35-s + 4.25·37-s + 4.28·39-s − 1.60·41-s + 4.57·43-s − 23.8·45-s + 10.4·47-s − 5.46·49-s + 3.05·51-s + 3.40·53-s + 23.5·55-s + ⋯ |
| L(s) = 1 | + 1.76·3-s − 1.68·5-s − 0.467·7-s + 2.10·9-s − 1.88·11-s + 0.389·13-s − 2.97·15-s + 0.242·17-s − 1.34·19-s − 0.824·21-s − 0.998·23-s + 1.85·25-s + 1.94·27-s + 0.671·29-s + 1.29·31-s − 3.31·33-s + 0.790·35-s + 0.699·37-s + 0.686·39-s − 0.250·41-s + 0.697·43-s − 3.55·45-s + 1.52·47-s − 0.781·49-s + 0.427·51-s + 0.468·53-s + 3.17·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.964160874\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.964160874\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 - 3.05T + 3T^{2} \) |
| 5 | \( 1 + 3.77T + 5T^{2} \) |
| 7 | \( 1 + 1.23T + 7T^{2} \) |
| 11 | \( 1 + 6.24T + 11T^{2} \) |
| 13 | \( 1 - 1.40T + 13T^{2} \) |
| 19 | \( 1 + 5.85T + 19T^{2} \) |
| 23 | \( 1 + 4.79T + 23T^{2} \) |
| 29 | \( 1 - 3.61T + 29T^{2} \) |
| 31 | \( 1 - 7.20T + 31T^{2} \) |
| 37 | \( 1 - 4.25T + 37T^{2} \) |
| 41 | \( 1 + 1.60T + 41T^{2} \) |
| 43 | \( 1 - 4.57T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 3.40T + 53T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 + 3.16T + 67T^{2} \) |
| 71 | \( 1 - 3.51T + 71T^{2} \) |
| 73 | \( 1 + 15.4T + 73T^{2} \) |
| 79 | \( 1 - 4.98T + 79T^{2} \) |
| 83 | \( 1 + 1.57T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 + 1.93T + 97T^{2} \) |
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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
−8.002445284809522821139934423465, −7.48088752936423190118903476072, −6.81126012447449055397061445389, −5.78566902287829038245119140821, −4.46457071539236518275222919464, −4.26601987947392789512218819285, −3.35577946807080039984601097916, −2.82723317355956189640281546361, −2.18092368062100535755025777895, −0.60241828230597177657141010298,
0.60241828230597177657141010298, 2.18092368062100535755025777895, 2.82723317355956189640281546361, 3.35577946807080039984601097916, 4.26601987947392789512218819285, 4.46457071539236518275222919464, 5.78566902287829038245119140821, 6.81126012447449055397061445389, 7.48088752936423190118903476072, 8.002445284809522821139934423465
