L(s) = 1 | + 2.48·2-s − 3-s + 4.16·4-s − 5-s − 2.48·6-s − 7-s + 5.37·8-s + 9-s − 2.48·10-s − 4.49·11-s − 4.16·12-s + 3.31·13-s − 2.48·14-s + 15-s + 5.02·16-s + 6.44·17-s + 2.48·18-s + 5.60·19-s − 4.16·20-s + 21-s − 11.1·22-s + 23-s − 5.37·24-s + 25-s + 8.22·26-s − 27-s − 4.16·28-s + ⋯ |
L(s) = 1 | + 1.75·2-s − 0.577·3-s + 2.08·4-s − 0.447·5-s − 1.01·6-s − 0.377·7-s + 1.90·8-s + 0.333·9-s − 0.785·10-s − 1.35·11-s − 1.20·12-s + 0.918·13-s − 0.663·14-s + 0.258·15-s + 1.25·16-s + 1.56·17-s + 0.585·18-s + 1.28·19-s − 0.931·20-s + 0.218·21-s − 2.38·22-s + 0.208·23-s − 1.09·24-s + 0.200·25-s + 1.61·26-s − 0.192·27-s − 0.787·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.092745853\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.092745853\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 - 2.48T + 2T^{2} \) |
| 11 | \( 1 + 4.49T + 11T^{2} \) |
| 13 | \( 1 - 3.31T + 13T^{2} \) |
| 17 | \( 1 - 6.44T + 17T^{2} \) |
| 19 | \( 1 - 5.60T + 19T^{2} \) |
| 29 | \( 1 - 7.13T + 29T^{2} \) |
| 31 | \( 1 - 6.09T + 31T^{2} \) |
| 37 | \( 1 + 5.01T + 37T^{2} \) |
| 41 | \( 1 - 3.10T + 41T^{2} \) |
| 43 | \( 1 - 4.56T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 - 9.27T + 53T^{2} \) |
| 59 | \( 1 + 5.59T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 - 0.375T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 - 3.75T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 + 2.50T + 97T^{2} \) |
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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
−8.867186219067898867400784245811, −7.67516576652109216984242567546, −7.32944519752651132613261962129, −6.19388333494425523536870291449, −5.71124101144797237263041389450, −5.04329366067042279487903958490, −4.25578055879879623099008729996, −3.26149323535511773965572093661, −2.78834143081207331667366058893, −1.08190853195154388967198164855,
1.08190853195154388967198164855, 2.78834143081207331667366058893, 3.26149323535511773965572093661, 4.25578055879879623099008729996, 5.04329366067042279487903958490, 5.71124101144797237263041389450, 6.19388333494425523536870291449, 7.32944519752651132613261962129, 7.67516576652109216984242567546, 8.867186219067898867400784245811