L(s) = 1 | + 2-s + 4-s + 5-s − 3.96·7-s + 8-s + 10-s − 3.04·11-s + 0.919·13-s − 3.96·14-s + 16-s − 0.351·17-s − 0.351·19-s + 20-s − 3.04·22-s + 3.43·23-s + 25-s + 0.919·26-s − 3.96·28-s − 9.37·29-s + 2.91·31-s + 32-s − 0.351·34-s − 3.96·35-s + 5.40·37-s − 0.351·38-s + 40-s + 8.45·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.447·5-s − 1.50·7-s + 0.353·8-s + 0.316·10-s − 0.919·11-s + 0.254·13-s − 1.06·14-s + 0.250·16-s − 0.0853·17-s − 0.0807·19-s + 0.223·20-s − 0.650·22-s + 0.715·23-s + 0.200·25-s + 0.180·26-s − 0.750·28-s − 1.74·29-s + 0.524·31-s + 0.176·32-s − 0.0603·34-s − 0.670·35-s + 0.888·37-s − 0.0570·38-s + 0.158·40-s + 1.31·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.562739145\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.562739145\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + T \) |
good | 7 | \( 1 + 3.96T + 7T^{2} \) |
| 11 | \( 1 + 3.04T + 11T^{2} \) |
| 13 | \( 1 - 0.919T + 13T^{2} \) |
| 17 | \( 1 + 0.351T + 17T^{2} \) |
| 19 | \( 1 + 0.351T + 19T^{2} \) |
| 23 | \( 1 - 3.43T + 23T^{2} \) |
| 29 | \( 1 + 9.37T + 29T^{2} \) |
| 31 | \( 1 - 2.91T + 31T^{2} \) |
| 37 | \( 1 - 5.40T + 37T^{2} \) |
| 41 | \( 1 - 8.45T + 41T^{2} \) |
| 43 | \( 1 - 6.09T + 43T^{2} \) |
| 47 | \( 1 - 13.2T + 47T^{2} \) |
| 53 | \( 1 + 5.93T + 53T^{2} \) |
| 59 | \( 1 - 8.45T + 59T^{2} \) |
| 61 | \( 1 + 1.96T + 61T^{2} \) |
| 71 | \( 1 - 5.61T + 71T^{2} \) |
| 73 | \( 1 + 12.8T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 + 0.105T + 83T^{2} \) |
| 89 | \( 1 - 16.2T + 89T^{2} \) |
| 97 | \( 1 - 3.04T + 97T^{2} \) |
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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
−7.79196754163964758584457197370, −7.29792573233465699145582275364, −6.44937470681751327200263546527, −5.91063647297601553801023757949, −5.38299390647734274583868840495, −4.37912011577070927089298769787, −3.60768236894160735178602150946, −2.82408507323631665595631495747, −2.23041256326920294202043202255, −0.72780526784434483781654196629,
0.72780526784434483781654196629, 2.23041256326920294202043202255, 2.82408507323631665595631495747, 3.60768236894160735178602150946, 4.37912011577070927089298769787, 5.38299390647734274583868840495, 5.91063647297601553801023757949, 6.44937470681751327200263546527, 7.29792573233465699145582275364, 7.79196754163964758584457197370