L(s) = 1 | + 2-s − 3.11·3-s + 4-s − 5-s − 3.11·6-s + 4.50·7-s + 8-s + 6.72·9-s − 10-s + 4.33·11-s − 3.11·12-s − 3.72·13-s + 4.50·14-s + 3.11·15-s + 16-s + 1.11·17-s + 6.72·18-s + 4.50·19-s − 20-s − 14.0·21-s + 4.33·22-s − 23-s − 3.11·24-s + 25-s − 3.72·26-s − 11.6·27-s + 4.50·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.80·3-s + 0.5·4-s − 0.447·5-s − 1.27·6-s + 1.70·7-s + 0.353·8-s + 2.24·9-s − 0.316·10-s + 1.30·11-s − 0.900·12-s − 1.03·13-s + 1.20·14-s + 0.805·15-s + 0.250·16-s + 0.271·17-s + 1.58·18-s + 1.03·19-s − 0.223·20-s − 3.06·21-s + 0.924·22-s − 0.208·23-s − 0.636·24-s + 0.200·25-s − 0.731·26-s − 2.23·27-s + 0.852·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.234852892\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.234852892\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 + 3.11T + 3T^{2} \) |
| 7 | \( 1 - 4.50T + 7T^{2} \) |
| 11 | \( 1 - 4.33T + 11T^{2} \) |
| 13 | \( 1 + 3.72T + 13T^{2} \) |
| 17 | \( 1 - 1.11T + 17T^{2} \) |
| 19 | \( 1 - 4.50T + 19T^{2} \) |
| 29 | \( 1 + 8.23T + 29T^{2} \) |
| 31 | \( 1 - 1.72T + 31T^{2} \) |
| 37 | \( 1 + 0.781T + 37T^{2} \) |
| 41 | \( 1 - 3.90T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 2.23T + 59T^{2} \) |
| 61 | \( 1 - 3.55T + 61T^{2} \) |
| 67 | \( 1 - 2.43T + 67T^{2} \) |
| 71 | \( 1 - 7.11T + 71T^{2} \) |
| 73 | \( 1 + 9.45T + 73T^{2} \) |
| 79 | \( 1 + 14.9T + 79T^{2} \) |
| 83 | \( 1 - 2.78T + 83T^{2} \) |
| 89 | \( 1 + 7.69T + 89T^{2} \) |
| 97 | \( 1 + 0.642T + 97T^{2} \) |
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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
−11.84184358877004945878526431087, −11.56894913203926316660027041159, −10.87554970774962297917492091612, −9.597989922901427020940416258005, −7.77235485781908469256542555775, −6.99661740520491590900847702223, −5.73226717343895512702744999918, −4.93968756083985667798027338432, −4.13461822791880196358785288925, −1.43901946297639245435214581985,
1.43901946297639245435214581985, 4.13461822791880196358785288925, 4.93968756083985667798027338432, 5.73226717343895512702744999918, 6.99661740520491590900847702223, 7.77235485781908469256542555775, 9.597989922901427020940416258005, 10.87554970774962297917492091612, 11.56894913203926316660027041159, 11.84184358877004945878526431087