L(s) = 1 | + 2·7-s − 13-s − 4·17-s + 6·19-s + 6·23-s − 4·29-s + 8·31-s + 6·37-s − 6·41-s − 4·43-s + 8·47-s − 3·49-s + 2·53-s − 2·61-s + 4·67-s + 8·71-s + 16·79-s − 4·83-s + 6·89-s − 2·91-s − 16·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 8·119-s + ⋯ |
L(s) = 1 | + 0.755·7-s − 0.277·13-s − 0.970·17-s + 1.37·19-s + 1.25·23-s − 0.742·29-s + 1.43·31-s + 0.986·37-s − 0.937·41-s − 0.609·43-s + 1.16·47-s − 3/7·49-s + 0.274·53-s − 0.256·61-s + 0.488·67-s + 0.949·71-s + 1.80·79-s − 0.439·83-s + 0.635·89-s − 0.209·91-s − 1.62·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.733·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.624020311\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.624020311\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−15.33644749493599, −15.02658197868288, −14.40703747935694, −13.71931968855809, −13.46655009015386, −12.80070960287372, −12.07983500883149, −11.62079457157643, −11.15313098488005, −10.64711752835061, −9.904772777921337, −9.350824435119523, −8.877496795172039, −8.077548471134230, −7.763969819378442, −6.927080367448202, −6.580198119182986, −5.614454917845233, −5.074720548551004, −4.609284929446142, −3.824859018396337, −3.001540788915733, −2.364819464777020, −1.466699832147713, −0.6853873531279163,
0.6853873531279163, 1.466699832147713, 2.364819464777020, 3.001540788915733, 3.824859018396337, 4.609284929446142, 5.074720548551004, 5.614454917845233, 6.580198119182986, 6.927080367448202, 7.763969819378442, 8.077548471134230, 8.877496795172039, 9.350824435119523, 9.904772777921337, 10.64711752835061, 11.15313098488005, 11.62079457157643, 12.07983500883149, 12.80070960287372, 13.46655009015386, 13.71931968855809, 14.40703747935694, 15.02658197868288, 15.33644749493599