L(s) = 1 | + 2-s − 3-s + 4-s + 1.41·5-s − 6-s + 7-s + 8-s + 9-s + 1.41·10-s − 4.07·11-s − 12-s + 14-s − 1.41·15-s + 16-s + 0.175·17-s + 18-s + 3.70·19-s + 1.41·20-s − 21-s − 4.07·22-s − 7.64·23-s − 24-s − 2.99·25-s − 27-s + 28-s + 8.67·29-s − 1.41·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.632·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.447·10-s − 1.22·11-s − 0.288·12-s + 0.267·14-s − 0.365·15-s + 0.250·16-s + 0.0425·17-s + 0.235·18-s + 0.851·19-s + 0.316·20-s − 0.218·21-s − 0.868·22-s − 1.59·23-s − 0.204·24-s − 0.599·25-s − 0.192·27-s + 0.188·28-s + 1.61·29-s − 0.258·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.911585953\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.911585953\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 1.41T + 5T^{2} \) |
| 11 | \( 1 + 4.07T + 11T^{2} \) |
| 17 | \( 1 - 0.175T + 17T^{2} \) |
| 19 | \( 1 - 3.70T + 19T^{2} \) |
| 23 | \( 1 + 7.64T + 23T^{2} \) |
| 29 | \( 1 - 8.67T + 29T^{2} \) |
| 31 | \( 1 + 10.8T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 + 0.0778T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 - 6.10T + 47T^{2} \) |
| 53 | \( 1 - 1.46T + 53T^{2} \) |
| 59 | \( 1 - 2.37T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 - 0.967T + 71T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 + 6.50T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−7.73284545929719775253769230720, −7.23732626439553436724073911188, −6.24039210083327166922533156221, −5.66648382769838115514050167309, −5.32586588133571823638275289960, −4.44672127597338110611268122229, −3.74606878944440998710775980102, −2.57678245278914176051160984915, −2.04898031224833165662770888416, −0.794038398271018060505560825514,
0.794038398271018060505560825514, 2.04898031224833165662770888416, 2.57678245278914176051160984915, 3.74606878944440998710775980102, 4.44672127597338110611268122229, 5.32586588133571823638275289960, 5.66648382769838115514050167309, 6.24039210083327166922533156221, 7.23732626439553436724073911188, 7.73284545929719775253769230720