| L(s) = 1 | + 2-s + 0.515·3-s + 4-s + 3.68·5-s + 0.515·6-s + 8-s − 2.73·9-s + 3.68·10-s + 6.45·11-s + 0.515·12-s + 2.84·13-s + 1.89·15-s + 16-s − 1.53·17-s − 2.73·18-s − 3.91·19-s + 3.68·20-s + 6.45·22-s + 0.380·23-s + 0.515·24-s + 8.59·25-s + 2.84·26-s − 2.95·27-s − 29-s + 1.89·30-s + 7.98·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.297·3-s + 0.5·4-s + 1.64·5-s + 0.210·6-s + 0.353·8-s − 0.911·9-s + 1.16·10-s + 1.94·11-s + 0.148·12-s + 0.790·13-s + 0.490·15-s + 0.250·16-s − 0.372·17-s − 0.644·18-s − 0.899·19-s + 0.824·20-s + 1.37·22-s + 0.0794·23-s + 0.105·24-s + 1.71·25-s + 0.558·26-s − 0.568·27-s − 0.185·29-s + 0.346·30-s + 1.43·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.661243607\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.661243607\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 0.515T + 3T^{2} \) |
| 5 | \( 1 - 3.68T + 5T^{2} \) |
| 11 | \( 1 - 6.45T + 11T^{2} \) |
| 13 | \( 1 - 2.84T + 13T^{2} \) |
| 17 | \( 1 + 1.53T + 17T^{2} \) |
| 19 | \( 1 + 3.91T + 19T^{2} \) |
| 23 | \( 1 - 0.380T + 23T^{2} \) |
| 31 | \( 1 - 7.98T + 31T^{2} \) |
| 37 | \( 1 + 7.46T + 37T^{2} \) |
| 41 | \( 1 + 4.98T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 - 7.85T + 47T^{2} \) |
| 53 | \( 1 + 1.22T + 53T^{2} \) |
| 59 | \( 1 + 0.696T + 59T^{2} \) |
| 61 | \( 1 + 3.10T + 61T^{2} \) |
| 67 | \( 1 + 7.22T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 - 14.9T + 73T^{2} \) |
| 79 | \( 1 - 5.13T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 - 9.39T + 89T^{2} \) |
| 97 | \( 1 + 2.56T + 97T^{2} \) |
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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
−8.903270715080827389574339068396, −8.254730213285490191753250860101, −6.67786904682324291180799112045, −6.50094705074529478546367927016, −5.83223597957145534155167627665, −4.98361628759528700619066551467, −3.97474853440435854063265531179, −3.13458134669840016615305862346, −2.11208060631243632324919051219, −1.38825761617845043623551612343,
1.38825761617845043623551612343, 2.11208060631243632324919051219, 3.13458134669840016615305862346, 3.97474853440435854063265531179, 4.98361628759528700619066551467, 5.83223597957145534155167627665, 6.50094705074529478546367927016, 6.67786904682324291180799112045, 8.254730213285490191753250860101, 8.903270715080827389574339068396
