| L(s) = 1 | − 0.185·2-s − 1.44·3-s − 1.96·4-s − 2.00·5-s + 0.268·6-s + 0.596·7-s + 0.734·8-s − 0.903·9-s + 0.371·10-s − 11-s + 2.84·12-s − 2.18·13-s − 0.110·14-s + 2.90·15-s + 3.79·16-s + 0.167·18-s − 8.53·19-s + 3.93·20-s − 0.863·21-s + 0.185·22-s − 0.641·23-s − 1.06·24-s − 0.982·25-s + 0.404·26-s + 5.65·27-s − 1.17·28-s − 8.45·29-s + ⋯ |
| L(s) = 1 | − 0.131·2-s − 0.836·3-s − 0.982·4-s − 0.896·5-s + 0.109·6-s + 0.225·7-s + 0.259·8-s − 0.301·9-s + 0.117·10-s − 0.301·11-s + 0.821·12-s − 0.604·13-s − 0.0295·14-s + 0.749·15-s + 0.948·16-s + 0.0394·18-s − 1.95·19-s + 0.880·20-s − 0.188·21-s + 0.0395·22-s − 0.133·23-s − 0.217·24-s − 0.196·25-s + 0.0792·26-s + 1.08·27-s − 0.221·28-s − 1.57·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3179 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3179 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.04686418301\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04686418301\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 0.185T + 2T^{2} \) |
| 3 | \( 1 + 1.44T + 3T^{2} \) |
| 5 | \( 1 + 2.00T + 5T^{2} \) |
| 7 | \( 1 - 0.596T + 7T^{2} \) |
| 13 | \( 1 + 2.18T + 13T^{2} \) |
| 19 | \( 1 + 8.53T + 19T^{2} \) |
| 23 | \( 1 + 0.641T + 23T^{2} \) |
| 29 | \( 1 + 8.45T + 29T^{2} \) |
| 31 | \( 1 + 9.08T + 31T^{2} \) |
| 37 | \( 1 - 3.56T + 37T^{2} \) |
| 41 | \( 1 - 5.76T + 41T^{2} \) |
| 43 | \( 1 + 3.06T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 + 3.74T + 53T^{2} \) |
| 59 | \( 1 + 6.20T + 59T^{2} \) |
| 61 | \( 1 + 0.0152T + 61T^{2} \) |
| 67 | \( 1 + 6.00T + 67T^{2} \) |
| 71 | \( 1 - 5.01T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + 9.76T + 79T^{2} \) |
| 83 | \( 1 - 9.28T + 83T^{2} \) |
| 89 | \( 1 + 16.2T + 89T^{2} \) |
| 97 | \( 1 - 13.9T + 97T^{2} \) |
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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
−8.617320388830167069893610749462, −7.955295350741141993474541226042, −7.36739848110623738786364639996, −6.25539700831464156179645457282, −5.57069516478666869714183315024, −4.75224367516611714317648670604, −4.20466844088562275346913252468, −3.29314029514041891103751206720, −1.86267706125243562294359400073, −0.13415638430709271209837144683,
0.13415638430709271209837144683, 1.86267706125243562294359400073, 3.29314029514041891103751206720, 4.20466844088562275346913252468, 4.75224367516611714317648670604, 5.57069516478666869714183315024, 6.25539700831464156179645457282, 7.36739848110623738786364639996, 7.955295350741141993474541226042, 8.617320388830167069893610749462
