| L(s) = 1 | + 2.56·2-s − 2.35·3-s + 4.56·4-s + 3.68·5-s − 6.04·6-s + 6.56·8-s + 2.56·9-s + 9.43·10-s − 11-s − 10.7·12-s − 3.39·13-s − 8.68·15-s + 7.68·16-s + 1.32·17-s + 6.56·18-s + 4.71·19-s + 16.7·20-s − 2.56·22-s − 5.56·23-s − 15.4·24-s + 8.56·25-s − 8.68·26-s + 1.03·27-s + 2·29-s − 22.2·30-s − 5.00·31-s + 6.56·32-s + ⋯ |
| L(s) = 1 | + 1.81·2-s − 1.36·3-s + 2.28·4-s + 1.64·5-s − 2.46·6-s + 2.31·8-s + 0.853·9-s + 2.98·10-s − 0.301·11-s − 3.10·12-s − 0.940·13-s − 2.24·15-s + 1.92·16-s + 0.321·17-s + 1.54·18-s + 1.08·19-s + 3.75·20-s − 0.546·22-s − 1.15·23-s − 3.15·24-s + 1.71·25-s − 1.70·26-s + 0.198·27-s + 0.371·29-s − 4.06·30-s − 0.899·31-s + 1.15·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.351127776\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.351127776\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 2.56T + 2T^{2} \) |
| 3 | \( 1 + 2.35T + 3T^{2} \) |
| 5 | \( 1 - 3.68T + 5T^{2} \) |
| 13 | \( 1 + 3.39T + 13T^{2} \) |
| 17 | \( 1 - 1.32T + 17T^{2} \) |
| 19 | \( 1 - 4.71T + 19T^{2} \) |
| 23 | \( 1 + 5.56T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 5.00T + 31T^{2} \) |
| 37 | \( 1 - 4.43T + 37T^{2} \) |
| 41 | \( 1 - 1.32T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 - 1.32T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 + 4.68T + 67T^{2} \) |
| 71 | \( 1 - 3.80T + 71T^{2} \) |
| 73 | \( 1 - 3.97T + 73T^{2} \) |
| 79 | \( 1 - 4.87T + 79T^{2} \) |
| 83 | \( 1 + 2.64T + 83T^{2} \) |
| 89 | \( 1 - 15.1T + 89T^{2} \) |
| 97 | \( 1 + 17.8T + 97T^{2} \) |
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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
−11.06423320190159121346973672871, −10.29455729161643232868061685174, −9.546789471708872955247478395676, −7.53037313436760588538134294029, −6.49557831984136005703236407973, −5.89746741429511823053146807779, −5.30038163124568471289529661524, −4.66836188292005232544315180817, −3.01547980077897075026585626782, −1.80626915283662163761189698673,
1.80626915283662163761189698673, 3.01547980077897075026585626782, 4.66836188292005232544315180817, 5.30038163124568471289529661524, 5.89746741429511823053146807779, 6.49557831984136005703236407973, 7.53037313436760588538134294029, 9.546789471708872955247478395676, 10.29455729161643232868061685174, 11.06423320190159121346973672871
