| L(s) = 1 | + 1.59·2-s − 1.55·3-s + 0.528·4-s − 5-s − 2.47·6-s − 1.60·7-s − 2.33·8-s − 0.570·9-s − 1.59·10-s + 1.84·11-s − 0.823·12-s + 1.04·13-s − 2.54·14-s + 1.55·15-s − 4.77·16-s + 2.68·17-s − 0.907·18-s + 5.62·19-s − 0.528·20-s + 2.49·21-s + 2.93·22-s − 0.721·23-s + 3.64·24-s + 25-s + 1.65·26-s + 5.56·27-s − 0.846·28-s + ⋯ |
| L(s) = 1 | + 1.12·2-s − 0.899·3-s + 0.264·4-s − 0.447·5-s − 1.01·6-s − 0.605·7-s − 0.827·8-s − 0.190·9-s − 0.502·10-s + 0.555·11-s − 0.237·12-s + 0.289·13-s − 0.680·14-s + 0.402·15-s − 1.19·16-s + 0.652·17-s − 0.213·18-s + 1.29·19-s − 0.118·20-s + 0.544·21-s + 0.624·22-s − 0.150·23-s + 0.744·24-s + 0.200·25-s + 0.325·26-s + 1.07·27-s − 0.160·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 37 | \( 1 \) |
| good | 2 | \( 1 - 1.59T + 2T^{2} \) |
| 3 | \( 1 + 1.55T + 3T^{2} \) |
| 7 | \( 1 + 1.60T + 7T^{2} \) |
| 11 | \( 1 - 1.84T + 11T^{2} \) |
| 13 | \( 1 - 1.04T + 13T^{2} \) |
| 17 | \( 1 - 2.68T + 17T^{2} \) |
| 19 | \( 1 - 5.62T + 19T^{2} \) |
| 23 | \( 1 + 0.721T + 23T^{2} \) |
| 29 | \( 1 + 6.48T + 29T^{2} \) |
| 31 | \( 1 - 5.95T + 31T^{2} \) |
| 41 | \( 1 + 8.18T + 41T^{2} \) |
| 43 | \( 1 - 8.15T + 43T^{2} \) |
| 47 | \( 1 + 1.59T + 47T^{2} \) |
| 53 | \( 1 - 2.75T + 53T^{2} \) |
| 59 | \( 1 + 0.907T + 59T^{2} \) |
| 61 | \( 1 + 1.32T + 61T^{2} \) |
| 67 | \( 1 - 3.08T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 + 11.9T + 73T^{2} \) |
| 79 | \( 1 - 2.60T + 79T^{2} \) |
| 83 | \( 1 - 2.06T + 83T^{2} \) |
| 89 | \( 1 - 3.87T + 89T^{2} \) |
| 97 | \( 1 + 2.70T + 97T^{2} \) |
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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
−7.37342197113652002162913845875, −6.54733932664193599584220159334, −6.09510141615696291017815874413, −5.39078993307310357701513210395, −4.93371050175323177841753347509, −3.95210013147323619556854448634, −3.44200191648198958649445079130, −2.70662839328890433539137206657, −1.11663835895365173972008984924, 0,
1.11663835895365173972008984924, 2.70662839328890433539137206657, 3.44200191648198958649445079130, 3.95210013147323619556854448634, 4.93371050175323177841753347509, 5.39078993307310357701513210395, 6.09510141615696291017815874413, 6.54733932664193599584220159334, 7.37342197113652002162913845875
