L(s) = 1 | + 2.45·2-s + 3-s + 4.03·4-s − 5-s + 2.45·6-s + 3.28·7-s + 4.98·8-s + 9-s − 2.45·10-s + 4.03·12-s − 0.313·13-s + 8.07·14-s − 15-s + 4.18·16-s + 5·17-s + 2.45·18-s − 7.45·19-s − 4.03·20-s + 3.28·21-s + 1.07·23-s + 4.98·24-s + 25-s − 0.769·26-s + 27-s + 13.2·28-s − 5.03·29-s − 2.45·30-s + ⋯ |
L(s) = 1 | + 1.73·2-s + 0.577·3-s + 2.01·4-s − 0.447·5-s + 1.00·6-s + 1.24·7-s + 1.76·8-s + 0.333·9-s − 0.776·10-s + 1.16·12-s − 0.0868·13-s + 2.15·14-s − 0.258·15-s + 1.04·16-s + 1.21·17-s + 0.578·18-s − 1.71·19-s − 0.901·20-s + 0.717·21-s + 0.223·23-s + 1.01·24-s + 0.200·25-s − 0.150·26-s + 0.192·27-s + 2.50·28-s − 0.935·29-s − 0.448·30-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
\(L(1)\) |
\(\approx\) |
\(6.243022332\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.243022332\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.45T + 2T^{2} \) |
| 7 | \( 1 - 3.28T + 7T^{2} \) |
| 13 | \( 1 + 0.313T + 13T^{2} \) |
| 17 | \( 1 - 5T + 17T^{2} \) |
| 19 | \( 1 + 7.45T + 19T^{2} \) |
| 23 | \( 1 - 1.07T + 23T^{2} \) |
| 29 | \( 1 + 5.03T + 29T^{2} \) |
| 31 | \( 1 - 3.44T + 31T^{2} \) |
| 37 | \( 1 - 2.63T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 + 5.51T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 - 4.93T + 53T^{2} \) |
| 59 | \( 1 + 9.16T + 59T^{2} \) |
| 61 | \( 1 - 9.18T + 61T^{2} \) |
| 67 | \( 1 + 15.2T + 67T^{2} \) |
| 71 | \( 1 - 3.07T + 71T^{2} \) |
| 73 | \( 1 + 8.65T + 73T^{2} \) |
| 79 | \( 1 - 5.41T + 79T^{2} \) |
| 83 | \( 1 - 16.2T + 83T^{2} \) |
| 89 | \( 1 - 1.62T + 89T^{2} \) |
| 97 | \( 1 - 0.224T + 97T^{2} \) |
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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
−9.144297309508425177191124012247, −8.108746352872691356915768986002, −7.69920217382472075123932427649, −6.71104650920268404640329298172, −5.83897064653971665760027646975, −4.91710039791091740583652592908, −4.34804336993642581089328722107, −3.57336140780122485946257009545, −2.59075169639491749608767981169, −1.62169920211514867212014857156,
1.62169920211514867212014857156, 2.59075169639491749608767981169, 3.57336140780122485946257009545, 4.34804336993642581089328722107, 4.91710039791091740583652592908, 5.83897064653971665760027646975, 6.71104650920268404640329298172, 7.69920217382472075123932427649, 8.108746352872691356915768986002, 9.144297309508425177191124012247