L(s) = 1 | − 5.31·2-s + 20.2·4-s − 5·5-s − 13.4·7-s − 65.1·8-s + 26.5·10-s − 46.9·11-s − 36.1·13-s + 71.4·14-s + 184.·16-s − 54.6·17-s + 111.·19-s − 101.·20-s + 249.·22-s − 35.9·23-s + 25·25-s + 192.·26-s − 272.·28-s + 58.1·29-s − 295.·31-s − 458.·32-s + 290.·34-s + 67.1·35-s − 53.0·37-s − 591.·38-s + 325.·40-s + 128.·41-s + ⋯ |
L(s) = 1 | − 1.87·2-s + 2.53·4-s − 0.447·5-s − 0.725·7-s − 2.87·8-s + 0.840·10-s − 1.28·11-s − 0.770·13-s + 1.36·14-s + 2.87·16-s − 0.779·17-s + 1.34·19-s − 1.13·20-s + 2.41·22-s − 0.326·23-s + 0.200·25-s + 1.44·26-s − 1.83·28-s + 0.372·29-s − 1.71·31-s − 2.53·32-s + 1.46·34-s + 0.324·35-s − 0.235·37-s − 2.52·38-s + 1.28·40-s + 0.488·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.3006493716\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3006493716\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
good | 2 | \( 1 + 5.31T + 8T^{2} \) |
| 7 | \( 1 + 13.4T + 343T^{2} \) |
| 11 | \( 1 + 46.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 36.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 54.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 111.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 35.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 58.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 295.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 53.0T + 5.06e4T^{2} \) |
| 41 | \( 1 - 128.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 164.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 87.8T + 1.03e5T^{2} \) |
| 53 | \( 1 + 479.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 635.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 48.0T + 2.26e5T^{2} \) |
| 67 | \( 1 + 28.9T + 3.00e5T^{2} \) |
| 71 | \( 1 - 576.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 835.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 203.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 464.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 993.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 881.T + 9.12e5T^{2} \) |
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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
−10.58428834126319441926391763160, −9.781951395298130911868815767378, −9.146659809953380007680642174546, −8.047513787547059840775239215125, −7.45045003874848915357626070349, −6.61830455565005486472325099457, −5.28584523359216844802227047046, −3.25031630265613507115998125630, −2.17658793496912945396131150029, −0.44145974167335774490302015681,
0.44145974167335774490302015681, 2.17658793496912945396131150029, 3.25031630265613507115998125630, 5.28584523359216844802227047046, 6.61830455565005486472325099457, 7.45045003874848915357626070349, 8.047513787547059840775239215125, 9.146659809953380007680642174546, 9.781951395298130911868815767378, 10.58428834126319441926391763160