| L(s) = 1 | − 1.59·2-s − 2.69·3-s + 0.531·4-s − 1.39·5-s + 4.29·6-s − 3.40·7-s + 2.33·8-s + 4.27·9-s + 2.22·10-s + 3.00·11-s − 1.43·12-s + 4.01·13-s + 5.41·14-s + 3.77·15-s − 4.78·16-s − 5.59·17-s − 6.79·18-s + 4.72·19-s − 0.744·20-s + 9.17·21-s − 4.77·22-s − 2.45·23-s − 6.29·24-s − 3.04·25-s − 6.38·26-s − 3.43·27-s − 1.80·28-s + ⋯ |
| L(s) = 1 | − 1.12·2-s − 1.55·3-s + 0.265·4-s − 0.625·5-s + 1.75·6-s − 1.28·7-s + 0.825·8-s + 1.42·9-s + 0.703·10-s + 0.905·11-s − 0.414·12-s + 1.11·13-s + 1.44·14-s + 0.974·15-s − 1.19·16-s − 1.35·17-s − 1.60·18-s + 1.08·19-s − 0.166·20-s + 2.00·21-s − 1.01·22-s − 0.512·23-s − 1.28·24-s − 0.608·25-s − 1.25·26-s − 0.660·27-s − 0.341·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1841044139\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1841044139\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 + 1.59T + 2T^{2} \) |
| 3 | \( 1 + 2.69T + 3T^{2} \) |
| 5 | \( 1 + 1.39T + 5T^{2} \) |
| 7 | \( 1 + 3.40T + 7T^{2} \) |
| 11 | \( 1 - 3.00T + 11T^{2} \) |
| 13 | \( 1 - 4.01T + 13T^{2} \) |
| 17 | \( 1 + 5.59T + 17T^{2} \) |
| 19 | \( 1 - 4.72T + 19T^{2} \) |
| 23 | \( 1 + 2.45T + 23T^{2} \) |
| 29 | \( 1 - 0.534T + 29T^{2} \) |
| 31 | \( 1 + 8.47T + 31T^{2} \) |
| 37 | \( 1 + 4.64T + 37T^{2} \) |
| 41 | \( 1 + 2.25T + 41T^{2} \) |
| 47 | \( 1 + 0.893T + 47T^{2} \) |
| 53 | \( 1 + 9.68T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 - 2.23T + 61T^{2} \) |
| 67 | \( 1 - 3.27T + 67T^{2} \) |
| 71 | \( 1 + 7.07T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 - 2.34T + 79T^{2} \) |
| 83 | \( 1 - 3.50T + 83T^{2} \) |
| 89 | \( 1 - 0.915T + 89T^{2} \) |
| 97 | \( 1 - 1.91T + 97T^{2} \) |
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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
−9.263849350881431446362059544068, −8.707085939024767827081661241537, −7.58324313542458426278939033689, −6.79358410524733789212751433793, −6.33141319606414258904548250965, −5.39264382130374141784965048850, −4.25473283490041257921379248859, −3.56799637242002553922461657388, −1.57382141644233134752331209854, −0.38773112876772737362622982423,
0.38773112876772737362622982423, 1.57382141644233134752331209854, 3.56799637242002553922461657388, 4.25473283490041257921379248859, 5.39264382130374141784965048850, 6.33141319606414258904548250965, 6.79358410524733789212751433793, 7.58324313542458426278939033689, 8.707085939024767827081661241537, 9.263849350881431446362059544068
