| L(s) = 1 | − 2.71·2-s + 5.38·4-s + 1.67·5-s − 2.59·7-s − 9.18·8-s − 4.54·10-s − 0.199·13-s + 7.04·14-s + 14.1·16-s + 7.79·17-s − 1.13·19-s + 8.99·20-s + 1.79·23-s − 2.20·25-s + 0.541·26-s − 13.9·28-s − 4.02·29-s + 4.43·31-s − 20.1·32-s − 21.1·34-s − 4.33·35-s + 4.55·37-s + 3.07·38-s − 15.3·40-s + 4.49·41-s + 3.36·43-s − 4.88·46-s + ⋯ |
| L(s) = 1 | − 1.92·2-s + 2.69·4-s + 0.747·5-s − 0.979·7-s − 3.24·8-s − 1.43·10-s − 0.0552·13-s + 1.88·14-s + 3.54·16-s + 1.88·17-s − 0.260·19-s + 2.01·20-s + 0.374·23-s − 0.441·25-s + 0.106·26-s − 2.63·28-s − 0.747·29-s + 0.795·31-s − 3.57·32-s − 3.63·34-s − 0.732·35-s + 0.748·37-s + 0.499·38-s − 2.42·40-s + 0.702·41-s + 0.513·43-s − 0.720·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8685682270\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8685682270\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.71T + 2T^{2} \) |
| 5 | \( 1 - 1.67T + 5T^{2} \) |
| 7 | \( 1 + 2.59T + 7T^{2} \) |
| 13 | \( 1 + 0.199T + 13T^{2} \) |
| 17 | \( 1 - 7.79T + 17T^{2} \) |
| 19 | \( 1 + 1.13T + 19T^{2} \) |
| 23 | \( 1 - 1.79T + 23T^{2} \) |
| 29 | \( 1 + 4.02T + 29T^{2} \) |
| 31 | \( 1 - 4.43T + 31T^{2} \) |
| 37 | \( 1 - 4.55T + 37T^{2} \) |
| 41 | \( 1 - 4.49T + 41T^{2} \) |
| 43 | \( 1 - 3.36T + 43T^{2} \) |
| 47 | \( 1 - 8.35T + 47T^{2} \) |
| 53 | \( 1 - 0.660T + 53T^{2} \) |
| 59 | \( 1 - 4.53T + 59T^{2} \) |
| 61 | \( 1 - 0.455T + 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 - 11.9T + 79T^{2} \) |
| 83 | \( 1 + 2.89T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 + 4.72T + 97T^{2} \) |
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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
−7.70155183379294884147626627513, −7.28577198393943488726020772819, −6.40690640540634193946998835675, −5.99772737724208226075261741065, −5.38325081511648787694036842685, −3.84622025487085637607164542861, −2.97133264516127713190496604644, −2.39542718122711886623344910095, −1.40839489543712480808897334779, −0.62982365743652677146857723619,
0.62982365743652677146857723619, 1.40839489543712480808897334779, 2.39542718122711886623344910095, 2.97133264516127713190496604644, 3.84622025487085637607164542861, 5.38325081511648787694036842685, 5.99772737724208226075261741065, 6.40690640540634193946998835675, 7.28577198393943488726020772819, 7.70155183379294884147626627513
