L(s) = 1 | + 1.09·2-s + 3-s − 0.796·4-s + 0.842·5-s + 1.09·6-s + 0.628·7-s − 3.06·8-s + 9-s + 0.924·10-s − 2.80·11-s − 0.796·12-s + 13-s + 0.689·14-s + 0.842·15-s − 1.77·16-s − 3.27·17-s + 1.09·18-s + 6.74·19-s − 0.671·20-s + 0.628·21-s − 3.08·22-s − 8.54·23-s − 3.06·24-s − 4.28·25-s + 1.09·26-s + 27-s − 0.501·28-s + ⋯ |
L(s) = 1 | + 0.775·2-s + 0.577·3-s − 0.398·4-s + 0.376·5-s + 0.447·6-s + 0.237·7-s − 1.08·8-s + 0.333·9-s + 0.292·10-s − 0.846·11-s − 0.230·12-s + 0.277·13-s + 0.184·14-s + 0.217·15-s − 0.442·16-s − 0.794·17-s + 0.258·18-s + 1.54·19-s − 0.150·20-s + 0.137·21-s − 0.656·22-s − 1.78·23-s − 0.626·24-s − 0.857·25-s + 0.215·26-s + 0.192·27-s − 0.0946·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
good | 2 | \( 1 - 1.09T + 2T^{2} \) |
| 5 | \( 1 - 0.842T + 5T^{2} \) |
| 7 | \( 1 - 0.628T + 7T^{2} \) |
| 11 | \( 1 + 2.80T + 11T^{2} \) |
| 17 | \( 1 + 3.27T + 17T^{2} \) |
| 19 | \( 1 - 6.74T + 19T^{2} \) |
| 23 | \( 1 + 8.54T + 23T^{2} \) |
| 29 | \( 1 + 2.30T + 29T^{2} \) |
| 31 | \( 1 + 9.72T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 - 1.75T + 41T^{2} \) |
| 43 | \( 1 - 4.10T + 43T^{2} \) |
| 47 | \( 1 - 8.70T + 47T^{2} \) |
| 53 | \( 1 - 7.28T + 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 - 6.50T + 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 + 13.0T + 73T^{2} \) |
| 79 | \( 1 - 2.83T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 - 11.0T + 89T^{2} \) |
| 97 | \( 1 + 0.970T + 97T^{2} \) |
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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
−7.951933815210881526659726378639, −7.53565040614828124166622415641, −6.42776517060193368560937089000, −5.52361982459522729717538314958, −5.24232202064815333203322245930, −4.08348705287834521366200785283, −3.62894800552581346116318955861, −2.61095381277341803859125430128, −1.74364722395517435576520300517, 0,
1.74364722395517435576520300517, 2.61095381277341803859125430128, 3.62894800552581346116318955861, 4.08348705287834521366200785283, 5.24232202064815333203322245930, 5.52361982459522729717538314958, 6.42776517060193368560937089000, 7.53565040614828124166622415641, 7.951933815210881526659726378639