| L(s) = 1 | − 0.241·2-s − 0.709·3-s − 0.941·4-s − 5-s + 0.170·6-s − 1.77·7-s + 0.468·8-s − 0.497·9-s + 0.241·10-s − 1.13·11-s + 0.667·12-s + 13-s + 0.426·14-s + 0.709·15-s + 0.829·16-s − 1.94·17-s + 0.119·18-s + 0.941·20-s + 1.25·21-s + 0.273·22-s − 1.49·23-s − 0.332·24-s + 25-s − 0.241·26-s + 1.06·27-s + 1.66·28-s − 0.170·30-s + ⋯ |
| L(s) = 1 | − 0.241·2-s − 0.709·3-s − 0.941·4-s − 5-s + 0.170·6-s − 1.77·7-s + 0.468·8-s − 0.497·9-s + 0.241·10-s − 1.13·11-s + 0.667·12-s + 13-s + 0.426·14-s + 0.709·15-s + 0.829·16-s − 1.94·17-s + 0.119·18-s + 0.941·20-s + 1.25·21-s + 0.273·22-s − 1.49·23-s − 0.332·24-s + 25-s − 0.241·26-s + 1.06·27-s + 1.66·28-s − 0.170·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1264350806\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1264350806\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 0.241T + T^{2} \) |
| 3 | \( 1 + 0.709T + T^{2} \) |
| 7 | \( 1 + 1.77T + T^{2} \) |
| 11 | \( 1 + 1.13T + T^{2} \) |
| 17 | \( 1 + 1.94T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 1.49T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 0.241T + T^{2} \) |
| 47 | \( 1 - 1.49T + T^{2} \) |
| 53 | \( 1 - 1.13T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 1.13T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 1.77T + T^{2} \) |
| 97 | \( 1 - 0.709T + T^{2} \) |
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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
−9.083671320778104235246940932773, −8.770064816878816092053782621275, −7.88096340277525467115148366450, −6.94642634270792341765037601845, −6.11241867795303764982245389224, −5.46393505890614919354797273409, −4.30112016051150424567699631710, −3.73007221758697957997331910277, −2.66675529416695072123998211271, −0.35048341944976075604806210579,
0.35048341944976075604806210579, 2.66675529416695072123998211271, 3.73007221758697957997331910277, 4.30112016051150424567699631710, 5.46393505890614919354797273409, 6.11241867795303764982245389224, 6.94642634270792341765037601845, 7.88096340277525467115148366450, 8.770064816878816092053782621275, 9.083671320778104235246940932773
