| L(s) = 1 | − 3-s + 5-s + 9-s + 3·11-s − 4·13-s − 15-s + 4·19-s + 8·23-s − 4·25-s − 27-s − 3·29-s + 5·31-s − 3·33-s + 8·37-s + 4·39-s − 8·41-s + 6·43-s + 45-s − 10·47-s + 9·53-s + 3·55-s − 4·57-s + 5·59-s + 10·61-s − 4·65-s + 6·67-s − 8·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.904·11-s − 1.10·13-s − 0.258·15-s + 0.917·19-s + 1.66·23-s − 4/5·25-s − 0.192·27-s − 0.557·29-s + 0.898·31-s − 0.522·33-s + 1.31·37-s + 0.640·39-s − 1.24·41-s + 0.914·43-s + 0.149·45-s − 1.45·47-s + 1.23·53-s + 0.404·55-s − 0.529·57-s + 0.650·59-s + 1.28·61-s − 0.496·65-s + 0.733·67-s − 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.510055456\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.510055456\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−9.656316692800465470874692420928, −9.289247173120531501683312522143, −8.056121456321313725647783148994, −7.11626426149924993734455517023, −6.49357825739679976742967528543, −5.45659847882817544849677095726, −4.81024134267614863022340721983, −3.64063338825773430172991239116, −2.36584813170026975724470574418, −0.992681600840089495059545096944,
0.992681600840089495059545096944, 2.36584813170026975724470574418, 3.64063338825773430172991239116, 4.81024134267614863022340721983, 5.45659847882817544849677095726, 6.49357825739679976742967528543, 7.11626426149924993734455517023, 8.056121456321313725647783148994, 9.289247173120531501683312522143, 9.656316692800465470874692420928
