L(s) = 1 | + 2-s + 4-s + 7-s + 8-s + 2·11-s − 2.16·13-s + 14-s + 16-s − 5.32·17-s + 3·19-s + 2·22-s − 5.16·23-s − 2.16·26-s + 28-s + 8.16·29-s + 6.32·31-s + 32-s − 5.32·34-s − 11.4·37-s + 3·38-s + 4·41-s − 3.16·43-s + 2·44-s − 5.16·46-s + 8.48·47-s + 49-s − 2.16·52-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.377·7-s + 0.353·8-s + 0.603·11-s − 0.599·13-s + 0.267·14-s + 0.250·16-s − 1.29·17-s + 0.688·19-s + 0.426·22-s − 1.07·23-s − 0.424·26-s + 0.188·28-s + 1.51·29-s + 1.13·31-s + 0.176·32-s − 0.913·34-s − 1.88·37-s + 0.486·38-s + 0.624·41-s − 0.482·43-s + 0.301·44-s − 0.761·46-s + 1.23·47-s + 0.142·49-s − 0.299·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.529313320\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.529313320\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 2.16T + 13T^{2} \) |
| 17 | \( 1 + 5.32T + 17T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 23 | \( 1 + 5.16T + 23T^{2} \) |
| 29 | \( 1 - 8.16T + 29T^{2} \) |
| 31 | \( 1 - 6.32T + 31T^{2} \) |
| 37 | \( 1 + 11.4T + 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 + 3.16T + 43T^{2} \) |
| 47 | \( 1 - 8.48T + 47T^{2} \) |
| 53 | \( 1 - 6.16T + 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 - 15.4T + 67T^{2} \) |
| 71 | \( 1 - 2.32T + 71T^{2} \) |
| 73 | \( 1 - 8.32T + 73T^{2} \) |
| 79 | \( 1 - 4.48T + 79T^{2} \) |
| 83 | \( 1 + 3.16T + 83T^{2} \) |
| 89 | \( 1 - 5T + 89T^{2} \) |
| 97 | \( 1 - 7.48T + 97T^{2} \) |
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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
−7.57029916872560515994074720650, −6.80666350852720939488415435772, −6.44570425849597626422251060666, −5.52296296149824219893407750670, −4.87661083685185536396222788999, −4.27055051262382691251328816138, −3.58733371813895854385652933655, −2.57750537949978826207225020711, −1.97799695347965926190689532453, −0.806535013713941230570236808319,
0.806535013713941230570236808319, 1.97799695347965926190689532453, 2.57750537949978826207225020711, 3.58733371813895854385652933655, 4.27055051262382691251328816138, 4.87661083685185536396222788999, 5.52296296149824219893407750670, 6.44570425849597626422251060666, 6.80666350852720939488415435772, 7.57029916872560515994074720650