| L(s) = 1 | − 3.85i·3-s − 0.490·5-s − 5.87·9-s + 15.5i·11-s − 3.50·13-s + 1.89i·15-s + 24.1·17-s − 3.56i·19-s − 19.5i·23-s − 24.7·25-s − 12.0i·27-s − 10.9·29-s − 21.1i·31-s + 60.0·33-s + 58.4·37-s + ⋯ |
| L(s) = 1 | − 1.28i·3-s − 0.0980·5-s − 0.652·9-s + 1.41i·11-s − 0.269·13-s + 0.126i·15-s + 1.42·17-s − 0.187i·19-s − 0.851i·23-s − 0.990·25-s − 0.446i·27-s − 0.378·29-s − 0.683i·31-s + 1.81·33-s + 1.57·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.551645842\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.551645842\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 3.85iT - 9T^{2} \) |
| 5 | \( 1 + 0.490T + 25T^{2} \) |
| 11 | \( 1 - 15.5iT - 121T^{2} \) |
| 13 | \( 1 + 3.50T + 169T^{2} \) |
| 17 | \( 1 - 24.1T + 289T^{2} \) |
| 19 | \( 1 + 3.56iT - 361T^{2} \) |
| 23 | \( 1 + 19.5iT - 529T^{2} \) |
| 29 | \( 1 + 10.9T + 841T^{2} \) |
| 31 | \( 1 + 21.1iT - 961T^{2} \) |
| 37 | \( 1 - 58.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 54.1T + 1.68e3T^{2} \) |
| 43 | \( 1 + 35.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 64.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 87.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + 66.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 16.8T + 3.72e3T^{2} \) |
| 67 | \( 1 - 21.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 64.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 99.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 139. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 6.03iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 23.9T + 7.92e3T^{2} \) |
| 97 | \( 1 + 171.T + 9.40e3T^{2} \) |
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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
−8.805361466358411499097124603260, −7.76167535760569500423048609085, −7.49278475078332972926401855203, −6.68688218019007041296268836488, −5.84314097228806923863909604945, −4.84384322607277887020862618236, −3.79686332448051915283713636361, −2.43686548502093190838540004003, −1.71511378869093271327817503734, −0.46066890728053154967988001845,
1.18031852413092786729074137595, 2.95947413765420944415267702342, 3.60059775373264187224590112388, 4.42800511284128302207689952498, 5.52887133959320438214401902187, 5.88182328931666002897966341514, 7.28346336310059373891495612533, 8.091979129413519577505586293940, 8.856993366871925672827188725877, 9.771003957712534137849465033385
