| L(s) = 1 | − 5-s + 2.89·7-s − 3.43·11-s − 4.72·13-s − 2.61·17-s + 2.28·19-s + 23-s + 25-s − 2.53·29-s + 6.19·31-s − 2.89·35-s + 4.89·37-s − 10.4·41-s − 9.01·43-s + 2.28·47-s + 1.40·49-s − 0.682·53-s + 3.43·55-s + 14.2·59-s + 4.98·61-s + 4.72·65-s + 1.74·67-s + 6.32·71-s − 1.51·73-s − 9.94·77-s − 3.86·79-s + 4.61·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.09·7-s − 1.03·11-s − 1.31·13-s − 0.634·17-s + 0.523·19-s + 0.208·23-s + 0.200·25-s − 0.470·29-s + 1.11·31-s − 0.489·35-s + 0.805·37-s − 1.63·41-s − 1.37·43-s + 0.332·47-s + 0.200·49-s − 0.0936·53-s + 0.462·55-s + 1.85·59-s + 0.638·61-s + 0.586·65-s + 0.213·67-s + 0.751·71-s − 0.177·73-s − 1.13·77-s − 0.434·79-s + 0.506·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.505523889\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.505523889\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - 2.89T + 7T^{2} \) |
| 11 | \( 1 + 3.43T + 11T^{2} \) |
| 13 | \( 1 + 4.72T + 13T^{2} \) |
| 17 | \( 1 + 2.61T + 17T^{2} \) |
| 19 | \( 1 - 2.28T + 19T^{2} \) |
| 29 | \( 1 + 2.53T + 29T^{2} \) |
| 31 | \( 1 - 6.19T + 31T^{2} \) |
| 37 | \( 1 - 4.89T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 + 9.01T + 43T^{2} \) |
| 47 | \( 1 - 2.28T + 47T^{2} \) |
| 53 | \( 1 + 0.682T + 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 61 | \( 1 - 4.98T + 61T^{2} \) |
| 67 | \( 1 - 1.74T + 67T^{2} \) |
| 71 | \( 1 - 6.32T + 71T^{2} \) |
| 73 | \( 1 + 1.51T + 73T^{2} \) |
| 79 | \( 1 + 3.86T + 79T^{2} \) |
| 83 | \( 1 - 4.61T + 83T^{2} \) |
| 89 | \( 1 - 6.31T + 89T^{2} \) |
| 97 | \( 1 - 17.8T + 97T^{2} \) |
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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
−7.900665255305339178569143937760, −7.21941495570227934432946211200, −6.61014527591542051501988910941, −5.45250814979711014299741836143, −4.96183163639127106591683449240, −4.52336351759448119970158099903, −3.46263438954715637089913477911, −2.56848309304838944365447128978, −1.89097787696178421380325412060, −0.58641742493072488737702014572,
0.58641742493072488737702014572, 1.89097787696178421380325412060, 2.56848309304838944365447128978, 3.46263438954715637089913477911, 4.52336351759448119970158099903, 4.96183163639127106591683449240, 5.45250814979711014299741836143, 6.61014527591542051501988910941, 7.21941495570227934432946211200, 7.900665255305339178569143937760
