| L(s) = 1 | + 2.13·3-s − 1.12·5-s − 4.02·7-s + 1.54·9-s − 5.57·11-s + 0.335·13-s − 2.39·15-s + 4.79·17-s + 1.73·19-s − 8.58·21-s − 3.07·23-s − 3.74·25-s − 3.09·27-s + 6.66·29-s − 3.51·31-s − 11.8·33-s + 4.51·35-s + 5.13·37-s + 0.715·39-s + 4.64·41-s − 9.81·43-s − 1.73·45-s + 11.9·47-s + 9.20·49-s + 10.2·51-s + 11.4·53-s + 6.25·55-s + ⋯ |
| L(s) = 1 | + 1.23·3-s − 0.501·5-s − 1.52·7-s + 0.515·9-s − 1.67·11-s + 0.0930·13-s − 0.618·15-s + 1.16·17-s + 0.397·19-s − 1.87·21-s − 0.641·23-s − 0.748·25-s − 0.596·27-s + 1.23·29-s − 0.630·31-s − 2.06·33-s + 0.763·35-s + 0.844·37-s + 0.114·39-s + 0.725·41-s − 1.49·43-s − 0.258·45-s + 1.73·47-s + 1.31·49-s + 1.43·51-s + 1.57·53-s + 0.843·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.641789706\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.641789706\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 751 | \( 1 + T \) |
| good | 3 | \( 1 - 2.13T + 3T^{2} \) |
| 5 | \( 1 + 1.12T + 5T^{2} \) |
| 7 | \( 1 + 4.02T + 7T^{2} \) |
| 11 | \( 1 + 5.57T + 11T^{2} \) |
| 13 | \( 1 - 0.335T + 13T^{2} \) |
| 17 | \( 1 - 4.79T + 17T^{2} \) |
| 19 | \( 1 - 1.73T + 19T^{2} \) |
| 23 | \( 1 + 3.07T + 23T^{2} \) |
| 29 | \( 1 - 6.66T + 29T^{2} \) |
| 31 | \( 1 + 3.51T + 31T^{2} \) |
| 37 | \( 1 - 5.13T + 37T^{2} \) |
| 41 | \( 1 - 4.64T + 41T^{2} \) |
| 43 | \( 1 + 9.81T + 43T^{2} \) |
| 47 | \( 1 - 11.9T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 - 2.35T + 59T^{2} \) |
| 61 | \( 1 - 9.55T + 61T^{2} \) |
| 67 | \( 1 + 2.78T + 67T^{2} \) |
| 71 | \( 1 - 7.12T + 71T^{2} \) |
| 73 | \( 1 - 6.65T + 73T^{2} \) |
| 79 | \( 1 + 8.62T + 79T^{2} \) |
| 83 | \( 1 + 1.96T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 - 5.97T + 97T^{2} \) |
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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
−7.951120163870238061496879958801, −7.66809690654240144006804704322, −6.87055951703269685245243252571, −5.87198385194703990315672061792, −5.34072454614511773641796901583, −4.06717641190508177438372023644, −3.48468940752082730912133568481, −2.84510253744552249006995020791, −2.28926043117533003493671888488, −0.59525813923847470156447925414,
0.59525813923847470156447925414, 2.28926043117533003493671888488, 2.84510253744552249006995020791, 3.48468940752082730912133568481, 4.06717641190508177438372023644, 5.34072454614511773641796901583, 5.87198385194703990315672061792, 6.87055951703269685245243252571, 7.66809690654240144006804704322, 7.951120163870238061496879958801
