| L(s) = 1 | + (−2.21 − 0.311i)5-s − i·7-s − 2.21·11-s + 5.11i·13-s − 4.11i·17-s + 5.11·19-s + i·23-s + (4.80 + 1.37i)25-s − 2.93·29-s − 2.09·31-s + (−0.311 + 2.21i)35-s + 1.28i·37-s − 0.458·41-s − 2.14i·43-s + 2.40i·47-s + ⋯ |
| L(s) = 1 | + (−0.990 − 0.139i)5-s − 0.377i·7-s − 0.667·11-s + 1.41i·13-s − 0.998i·17-s + 1.17·19-s + 0.208i·23-s + (0.961 + 0.275i)25-s − 0.544·29-s − 0.376·31-s + (−0.0525 + 0.374i)35-s + 0.210i·37-s − 0.0716·41-s − 0.327i·43-s + 0.351i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8596100589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8596100589\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.21 + 0.311i)T \) |
| 23 | \( 1 - iT \) |
| good | 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 + 2.21T + 11T^{2} \) |
| 13 | \( 1 - 5.11iT - 13T^{2} \) |
| 17 | \( 1 + 4.11iT - 17T^{2} \) |
| 19 | \( 1 - 5.11T + 19T^{2} \) |
| 29 | \( 1 + 2.93T + 29T^{2} \) |
| 31 | \( 1 + 2.09T + 31T^{2} \) |
| 37 | \( 1 - 1.28iT - 37T^{2} \) |
| 41 | \( 1 + 0.458T + 41T^{2} \) |
| 43 | \( 1 + 2.14iT - 43T^{2} \) |
| 47 | \( 1 - 2.40iT - 47T^{2} \) |
| 53 | \( 1 - 3.83iT - 53T^{2} \) |
| 59 | \( 1 - 1.68T + 59T^{2} \) |
| 61 | \( 1 + 15.5T + 61T^{2} \) |
| 67 | \( 1 + 1.76iT - 67T^{2} \) |
| 71 | \( 1 + 4.39T + 71T^{2} \) |
| 73 | \( 1 + 11.7iT - 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 + 11.4iT - 83T^{2} \) |
| 89 | \( 1 - 0.755T + 89T^{2} \) |
| 97 | \( 1 + 11.8iT - 97T^{2} \) |
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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
−8.061540432560133974671465087185, −7.36181460719020103429922034505, −7.12242623386818724766875532600, −6.01559207482675058146379102379, −5.02639281840747823923151691793, −4.48705892754394761609861810163, −3.61650494142241932352894401443, −2.84086106192960037827762969078, −1.57670166144139205079791945104, −0.30258918360695199135918419396,
0.957246110538066770241751239158, 2.42111026150686823973823694975, 3.26667533450282591966305696936, 3.88365901049117031541945982207, 5.01904642349220435283882165428, 5.53553586268314715761823434130, 6.40132038324062714661130661977, 7.47241819076993418867610079844, 7.79103548343025402873825050751, 8.454857984701517030634443401497
