L(s) = 1 | + 3-s + 9-s − 2·11-s + 13-s + 8·17-s − 7·19-s + 6·23-s + 27-s + 4·29-s − 8·31-s − 2·33-s − 7·37-s + 39-s − 2·41-s − 4·43-s + 12·47-s + 8·51-s + 4·53-s − 7·57-s − 12·59-s − 3·61-s − 9·67-s + 6·69-s − 12·71-s + 9·73-s − 17·79-s + 81-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.603·11-s + 0.277·13-s + 1.94·17-s − 1.60·19-s + 1.25·23-s + 0.192·27-s + 0.742·29-s − 1.43·31-s − 0.348·33-s − 1.15·37-s + 0.160·39-s − 0.312·41-s − 0.609·43-s + 1.75·47-s + 1.12·51-s + 0.549·53-s − 0.927·57-s − 1.56·59-s − 0.384·61-s − 1.09·67-s + 0.722·69-s − 1.42·71-s + 1.05·73-s − 1.91·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + 17 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 11 T + p T^{2} \) |
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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
−15.23213049650338, −14.94475658538707, −14.42941554925489, −13.83513498237069, −13.37354292980906, −12.70947171516776, −12.39750660431971, −11.84058736853867, −10.93251792318186, −10.48010179053792, −10.24615943952999, −9.304620086889307, −8.933381275063855, −8.322208712240243, −7.830529827127246, −7.184467959886565, −6.742709856778752, −5.760536540040213, −5.459241122862947, −4.613139146780197, −3.984371601415935, −3.200437952712146, −2.822642873496331, −1.844847235525059, −1.189013528330304, 0,
1.189013528330304, 1.844847235525059, 2.822642873496331, 3.200437952712146, 3.984371601415935, 4.613139146780197, 5.459241122862947, 5.760536540040213, 6.742709856778752, 7.184467959886565, 7.830529827127246, 8.322208712240243, 8.933381275063855, 9.304620086889307, 10.24615943952999, 10.48010179053792, 10.93251792318186, 11.84058736853867, 12.39750660431971, 12.70947171516776, 13.37354292980906, 13.83513498237069, 14.42941554925489, 14.94475658538707, 15.23213049650338