| L(s) = 1 | − 3·5-s + 4·11-s − 13-s + 17-s + 4·23-s + 4·25-s + 29-s + 3·31-s + 2·37-s + 5·41-s + 6·43-s − 9·47-s + 2·53-s − 12·55-s − 59-s − 12·61-s + 3·65-s + 10·67-s + 8·73-s + 5·83-s − 3·85-s + 6·89-s + 12·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 1.20·11-s − 0.277·13-s + 0.242·17-s + 0.834·23-s + 4/5·25-s + 0.185·29-s + 0.538·31-s + 0.328·37-s + 0.780·41-s + 0.914·43-s − 1.31·47-s + 0.274·53-s − 1.61·55-s − 0.130·59-s − 1.53·61-s + 0.372·65-s + 1.22·67-s + 0.936·73-s + 0.548·83-s − 0.325·85-s + 0.635·89-s + 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29988 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29988 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.787630879\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.787630879\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−15.00383546613591, −14.75695485800430, −14.22135295162351, −13.58233535452926, −12.91638272193689, −12.24785765184672, −12.05590430922745, −11.39440221388327, −11.03329771384316, −10.42968573263102, −9.489685624958141, −9.312621958378128, −8.491916333462923, −8.017339105936953, −7.486602661894247, −6.888226284864712, −6.391913820257550, −5.648414135331871, −4.729205921573479, −4.402278047976074, −3.648861089865249, −3.217342986380655, −2.322882840105712, −1.262939526443419, −0.5743808226760613,
0.5743808226760613, 1.262939526443419, 2.322882840105712, 3.217342986380655, 3.648861089865249, 4.402278047976074, 4.729205921573479, 5.648414135331871, 6.391913820257550, 6.888226284864712, 7.486602661894247, 8.017339105936953, 8.491916333462923, 9.312621958378128, 9.489685624958141, 10.42968573263102, 11.03329771384316, 11.39440221388327, 12.05590430922745, 12.24785765184672, 12.91638272193689, 13.58233535452926, 14.22135295162351, 14.75695485800430, 15.00383546613591
