L(s) = 1 | + 2·3-s + 5-s − 3·7-s + 9-s + 11-s + 6·13-s + 2·15-s − 17-s + 6·19-s − 6·21-s − 4·23-s + 25-s − 4·27-s − 7·29-s − 2·31-s + 2·33-s − 3·35-s + 2·37-s + 12·39-s − 5·41-s + 10·43-s + 45-s + 7·47-s + 2·49-s − 2·51-s − 53-s + 55-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 0.447·5-s − 1.13·7-s + 1/3·9-s + 0.301·11-s + 1.66·13-s + 0.516·15-s − 0.242·17-s + 1.37·19-s − 1.30·21-s − 0.834·23-s + 1/5·25-s − 0.769·27-s − 1.29·29-s − 0.359·31-s + 0.348·33-s − 0.507·35-s + 0.328·37-s + 1.92·39-s − 0.780·41-s + 1.52·43-s + 0.149·45-s + 1.02·47-s + 2/7·49-s − 0.280·51-s − 0.137·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.769663991\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.769663991\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.35916569171666, −13.68699564922792, −13.37095821863133, −13.14403068135189, −12.41523506014240, −11.81376157164260, −11.21609918681879, −10.73008563994611, −9.992475506334928, −9.587855050401466, −9.119336199936884, −8.806618545090347, −8.214245276300104, −7.498194329278687, −7.161620506013827, −6.312045371334210, −5.855155860373016, −5.575600612767354, −4.415251569114317, −3.776960481902818, −3.400607876668066, −2.931154462541825, −2.127844533440088, −1.520585734318421, −0.6106416219362291,
0.6106416219362291, 1.520585734318421, 2.127844533440088, 2.931154462541825, 3.400607876668066, 3.776960481902818, 4.415251569114317, 5.575600612767354, 5.855155860373016, 6.312045371334210, 7.161620506013827, 7.498194329278687, 8.214245276300104, 8.806618545090347, 9.119336199936884, 9.587855050401466, 9.992475506334928, 10.73008563994611, 11.21609918681879, 11.81376157164260, 12.41523506014240, 13.14403068135189, 13.37095821863133, 13.68699564922792, 14.35916569171666