| L(s) = 1 | − 3-s + 9-s − 3·13-s + 2·17-s − 19-s − 2·23-s − 27-s − 8·29-s + 8·31-s + 7·37-s + 3·39-s + 8·43-s + 10·47-s − 2·51-s − 14·53-s + 57-s − 10·59-s + 7·61-s + 5·67-s + 2·69-s + 12·71-s − 11·73-s + 7·79-s + 81-s − 14·83-s + 8·87-s − 6·89-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.832·13-s + 0.485·17-s − 0.229·19-s − 0.417·23-s − 0.192·27-s − 1.48·29-s + 1.43·31-s + 1.15·37-s + 0.480·39-s + 1.21·43-s + 1.45·47-s − 0.280·51-s − 1.92·53-s + 0.132·57-s − 1.30·59-s + 0.896·61-s + 0.610·67-s + 0.240·69-s + 1.42·71-s − 1.28·73-s + 0.787·79-s + 1/9·81-s − 1.53·83-s + 0.857·87-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−14.48189544238302, −14.17758511633223, −13.60000587580395, −12.79176651498564, −12.68033983731666, −12.03979621329789, −11.55425215987856, −11.07361026680498, −10.53071591115266, −10.00639011650119, −9.453711961160465, −9.147447130736750, −8.212737754636891, −7.726320480767217, −7.403011387583762, −6.554885878268670, −6.211010177625064, −5.488940641998391, −5.131908567464859, −4.226550582006806, −4.066748471219875, −2.985698726116803, −2.486267164407312, −1.658362163140742, −0.8534932746752432, 0,
0.8534932746752432, 1.658362163140742, 2.486267164407312, 2.985698726116803, 4.066748471219875, 4.226550582006806, 5.131908567464859, 5.488940641998391, 6.211010177625064, 6.554885878268670, 7.403011387583762, 7.726320480767217, 8.212737754636891, 9.147447130736750, 9.453711961160465, 10.00639011650119, 10.53071591115266, 11.07361026680498, 11.55425215987856, 12.03979621329789, 12.68033983731666, 12.79176651498564, 13.60000587580395, 14.17758511633223, 14.48189544238302
