L(s) = 1 | + 2·3-s + 5-s + 9-s − 3·11-s + 4·13-s + 2·15-s + 2·17-s + 19-s − 7·23-s − 4·25-s − 4·27-s + 2·29-s + 6·31-s − 6·33-s − 10·37-s + 8·39-s + 8·41-s + 7·43-s + 45-s + 9·47-s + 4·51-s + 6·53-s − 3·55-s + 2·57-s + 14·59-s + 5·61-s + 4·65-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 0.447·5-s + 1/3·9-s − 0.904·11-s + 1.10·13-s + 0.516·15-s + 0.485·17-s + 0.229·19-s − 1.45·23-s − 4/5·25-s − 0.769·27-s + 0.371·29-s + 1.07·31-s − 1.04·33-s − 1.64·37-s + 1.28·39-s + 1.24·41-s + 1.06·43-s + 0.149·45-s + 1.31·47-s + 0.560·51-s + 0.824·53-s − 0.404·55-s + 0.264·57-s + 1.82·59-s + 0.640·61-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.374993715\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.374993715\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 17 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−8.052567334422649857826091515646, −7.46712766478967909353633740637, −6.47011709914738638224082363647, −5.76752663525708997195007098229, −5.22313585005548073268036613318, −3.92559668470188645679496190708, −3.64770193165967076405604715660, −2.48597725176336781972653187797, −2.17264391045067968392885390816, −0.870181391233899341603195972834,
0.870181391233899341603195972834, 2.17264391045067968392885390816, 2.48597725176336781972653187797, 3.64770193165967076405604715660, 3.92559668470188645679496190708, 5.22313585005548073268036613318, 5.76752663525708997195007098229, 6.47011709914738638224082363647, 7.46712766478967909353633740637, 8.052567334422649857826091515646