| L(s) = 1 | − 2·3-s + 3·7-s + 9-s + 3·11-s − 4·13-s − 5·17-s + 19-s − 6·21-s + 4·27-s − 2·29-s + 8·31-s − 6·33-s − 10·37-s + 8·39-s + 6·41-s − 7·43-s + 9·47-s + 2·49-s + 10·51-s − 8·53-s − 2·57-s − 14·59-s + 5·61-s + 3·63-s − 6·71-s + 15·73-s + 9·77-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.13·7-s + 1/3·9-s + 0.904·11-s − 1.10·13-s − 1.21·17-s + 0.229·19-s − 1.30·21-s + 0.769·27-s − 0.371·29-s + 1.43·31-s − 1.04·33-s − 1.64·37-s + 1.28·39-s + 0.937·41-s − 1.06·43-s + 1.31·47-s + 2/7·49-s + 1.40·51-s − 1.09·53-s − 0.264·57-s − 1.82·59-s + 0.640·61-s + 0.377·63-s − 0.712·71-s + 1.75·73-s + 1.02·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.152854774\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.152854774\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 15 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−15.29621412165612, −14.52009757723303, −14.05793201576142, −13.73863223393834, −12.75266320617648, −12.32140096402674, −11.76739659557094, −11.51921177053922, −10.92835250512927, −10.49846942621536, −9.823585043867921, −9.127037391228239, −8.655379124220430, −7.957545506948179, −7.340290770574687, −6.694372022030007, −6.297086750960572, −5.554256936819214, −4.798915204971667, −4.747892157319613, −3.921504885910710, −2.913314910155926, −2.079840150186839, −1.392322103750304, −0.4571297133377085,
0.4571297133377085, 1.392322103750304, 2.079840150186839, 2.913314910155926, 3.921504885910710, 4.747892157319613, 4.798915204971667, 5.554256936819214, 6.297086750960572, 6.694372022030007, 7.340290770574687, 7.957545506948179, 8.655379124220430, 9.127037391228239, 9.823585043867921, 10.49846942621536, 10.92835250512927, 11.51921177053922, 11.76739659557094, 12.32140096402674, 12.75266320617648, 13.73863223393834, 14.05793201576142, 14.52009757723303, 15.29621412165612
