| L(s) = 1 | + 2.13·2-s − 3-s + 2.55·4-s + 5-s − 2.13·6-s + 4.82·7-s + 1.17·8-s + 9-s + 2.13·10-s − 2.55·12-s + 4.26·13-s + 10.2·14-s − 15-s − 2.59·16-s + 4.64·17-s + 2.13·18-s − 6.37·19-s + 2.55·20-s − 4.82·21-s − 5.14·23-s − 1.17·24-s + 25-s + 9.10·26-s − 27-s + 12.3·28-s + 4.26·29-s − 2.13·30-s + ⋯ |
| L(s) = 1 | + 1.50·2-s − 0.577·3-s + 1.27·4-s + 0.447·5-s − 0.871·6-s + 1.82·7-s + 0.416·8-s + 0.333·9-s + 0.674·10-s − 0.736·12-s + 1.18·13-s + 2.74·14-s − 0.258·15-s − 0.647·16-s + 1.12·17-s + 0.502·18-s − 1.46·19-s + 0.570·20-s − 1.05·21-s − 1.07·23-s − 0.240·24-s + 0.200·25-s + 1.78·26-s − 0.192·27-s + 2.32·28-s + 0.792·29-s − 0.389·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.417050101\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.417050101\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.13T + 2T^{2} \) |
| 7 | \( 1 - 4.82T + 7T^{2} \) |
| 13 | \( 1 - 4.26T + 13T^{2} \) |
| 17 | \( 1 - 4.64T + 17T^{2} \) |
| 19 | \( 1 + 6.37T + 19T^{2} \) |
| 23 | \( 1 + 5.14T + 23T^{2} \) |
| 29 | \( 1 - 4.26T + 29T^{2} \) |
| 31 | \( 1 + 6.39T + 31T^{2} \) |
| 37 | \( 1 - 6.14T + 37T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 + 1.55T + 43T^{2} \) |
| 47 | \( 1 - 5.35T + 47T^{2} \) |
| 53 | \( 1 - 2.24T + 53T^{2} \) |
| 59 | \( 1 + 13.5T + 59T^{2} \) |
| 61 | \( 1 + 6.80T + 61T^{2} \) |
| 67 | \( 1 - 6.14T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 - 5.62T + 73T^{2} \) |
| 79 | \( 1 - 16.3T + 79T^{2} \) |
| 83 | \( 1 - 6.17T + 83T^{2} \) |
| 89 | \( 1 + 1.39T + 89T^{2} \) |
| 97 | \( 1 - 3.93T + 97T^{2} \) |
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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
−9.213814067811851485681567361032, −8.292521605289708476537047086662, −7.58640601461105568483844655993, −6.34744612025569780829202321591, −5.89308356864460289667698818999, −5.15893174156053473994377920823, −4.42336546718762866628302272030, −3.76898140057918541120645541967, −2.33039536642846141435141295528, −1.39463534143334385091833402099,
1.39463534143334385091833402099, 2.33039536642846141435141295528, 3.76898140057918541120645541967, 4.42336546718762866628302272030, 5.15893174156053473994377920823, 5.89308356864460289667698818999, 6.34744612025569780829202321591, 7.58640601461105568483844655993, 8.292521605289708476537047086662, 9.213814067811851485681567361032
