L(s) = 1 | + 3-s + 9-s + 6·11-s − 2·13-s − 4·19-s + 6·23-s − 5·25-s + 27-s + 6·29-s + 8·31-s + 6·33-s + 2·37-s − 2·39-s − 12·41-s + 4·43-s + 12·47-s − 6·53-s − 4·57-s + 10·61-s − 8·67-s + 6·69-s − 6·71-s + 10·73-s − 5·75-s + 4·79-s + 81-s − 12·83-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1/3·9-s + 1.80·11-s − 0.554·13-s − 0.917·19-s + 1.25·23-s − 25-s + 0.192·27-s + 1.11·29-s + 1.43·31-s + 1.04·33-s + 0.328·37-s − 0.320·39-s − 1.87·41-s + 0.609·43-s + 1.75·47-s − 0.824·53-s − 0.529·57-s + 1.28·61-s − 0.977·67-s + 0.722·69-s − 0.712·71-s + 1.17·73-s − 0.577·75-s + 0.450·79-s + 1/9·81-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.483586155\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.483586155\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−8.838743631265884733873712251525, −8.468053709441953264102830038362, −7.38551790922965845580297625528, −6.71059201271486274282279699706, −6.06943423181930863194996495192, −4.77020152406148631842736257259, −4.14682573537835113219312172425, −3.21811200375960790502858283729, −2.18760227953726616802940054530, −1.05217073555011886108482433001,
1.05217073555011886108482433001, 2.18760227953726616802940054530, 3.21811200375960790502858283729, 4.14682573537835113219312172425, 4.77020152406148631842736257259, 6.06943423181930863194996495192, 6.71059201271486274282279699706, 7.38551790922965845580297625528, 8.468053709441953264102830038362, 8.838743631265884733873712251525