| L(s) = 1 | − 1.68·2-s − 2.54·3-s + 0.827·4-s − 1.60·5-s + 4.28·6-s − 4.81·7-s + 1.97·8-s + 3.49·9-s + 2.69·10-s + 6.43·11-s − 2.10·12-s + 3.23·13-s + 8.09·14-s + 4.08·15-s − 4.97·16-s − 4.60·17-s − 5.88·18-s + 4.55·19-s − 1.32·20-s + 12.2·21-s − 10.8·22-s + 23-s − 5.02·24-s − 2.43·25-s − 5.44·26-s − 1.27·27-s − 3.98·28-s + ⋯ |
| L(s) = 1 | − 1.18·2-s − 1.47·3-s + 0.413·4-s − 0.715·5-s + 1.74·6-s − 1.81·7-s + 0.697·8-s + 1.16·9-s + 0.851·10-s + 1.94·11-s − 0.608·12-s + 0.897·13-s + 2.16·14-s + 1.05·15-s − 1.24·16-s − 1.11·17-s − 1.38·18-s + 1.04·19-s − 0.296·20-s + 2.67·21-s − 2.30·22-s + 0.208·23-s − 1.02·24-s − 0.487·25-s − 1.06·26-s − 0.245·27-s − 0.752·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 1.68T + 2T^{2} \) |
| 3 | \( 1 + 2.54T + 3T^{2} \) |
| 5 | \( 1 + 1.60T + 5T^{2} \) |
| 7 | \( 1 + 4.81T + 7T^{2} \) |
| 11 | \( 1 - 6.43T + 11T^{2} \) |
| 13 | \( 1 - 3.23T + 13T^{2} \) |
| 17 | \( 1 + 4.60T + 17T^{2} \) |
| 19 | \( 1 - 4.55T + 19T^{2} \) |
| 31 | \( 1 + 1.11T + 31T^{2} \) |
| 37 | \( 1 - 4.51T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 - 8.83T + 43T^{2} \) |
| 47 | \( 1 + 5.22T + 47T^{2} \) |
| 53 | \( 1 + 7.18T + 53T^{2} \) |
| 59 | \( 1 - 4.45T + 59T^{2} \) |
| 61 | \( 1 + 3.88T + 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 + 0.460T + 73T^{2} \) |
| 79 | \( 1 + 0.482T + 79T^{2} \) |
| 83 | \( 1 + 4.96T + 83T^{2} \) |
| 89 | \( 1 + 17.2T + 89T^{2} \) |
| 97 | \( 1 - 1.06T + 97T^{2} \) |
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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
−9.934489987577773026214514305322, −9.365590389366102633163182759982, −8.630824386938377471296604666522, −7.21340811516401463080767186361, −6.60825295922963666600838701940, −6.00314002731520815030442190596, −4.43007001086086446518466311095, −3.55540739124716878891278281454, −1.14250292755990052583986266435, 0,
1.14250292755990052583986266435, 3.55540739124716878891278281454, 4.43007001086086446518466311095, 6.00314002731520815030442190596, 6.60825295922963666600838701940, 7.21340811516401463080767186361, 8.630824386938377471296604666522, 9.365590389366102633163182759982, 9.934489987577773026214514305322
