| L(s) = 1 | − 3·5-s + 7-s − 3·9-s − 6·11-s + 4·17-s + 5·19-s + 3·23-s + 4·25-s + 5·29-s + 3·31-s − 3·35-s − 4·37-s + 6·41-s + 43-s + 9·45-s − 7·47-s + 49-s + 9·53-s + 18·55-s + 8·59-s + 10·61-s − 3·63-s − 6·67-s + 8·71-s + 13·73-s − 6·77-s + 3·79-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.377·7-s − 9-s − 1.80·11-s + 0.970·17-s + 1.14·19-s + 0.625·23-s + 4/5·25-s + 0.928·29-s + 0.538·31-s − 0.507·35-s − 0.657·37-s + 0.937·41-s + 0.152·43-s + 1.34·45-s − 1.02·47-s + 1/7·49-s + 1.23·53-s + 2.42·55-s + 1.04·59-s + 1.28·61-s − 0.377·63-s − 0.733·67-s + 0.949·71-s + 1.52·73-s − 0.683·77-s + 0.337·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.528631505\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.528631505\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−14.15742457693516, −13.51669598927238, −13.12439639452640, −12.30599567394755, −12.11183598857363, −11.56762310425578, −11.09172303563941, −10.70397945675504, −10.08522947619452, −9.600393178910016, −8.743745919740024, −8.285258489721680, −7.987711122068368, −7.536593886414390, −7.073792689694654, −6.263912882295114, −5.454013452909762, −5.182991232372382, −4.752034693059854, −3.797306390004491, −3.361132962237878, −2.771314375167276, −2.284407822606719, −0.9670104416967863, −0.5069344569135427,
0.5069344569135427, 0.9670104416967863, 2.284407822606719, 2.771314375167276, 3.361132962237878, 3.797306390004491, 4.752034693059854, 5.182991232372382, 5.454013452909762, 6.263912882295114, 7.073792689694654, 7.536593886414390, 7.987711122068368, 8.285258489721680, 8.743745919740024, 9.600393178910016, 10.08522947619452, 10.70397945675504, 11.09172303563941, 11.56762310425578, 12.11183598857363, 12.30599567394755, 13.12439639452640, 13.51669598927238, 14.15742457693516
