L(s) = 1 | + 1.90·2-s + 1.61·4-s − 5-s + 7-s − 0.733·8-s − 1.90·10-s − 3.08·11-s − 4.36·13-s + 1.90·14-s − 4.62·16-s + 3.91·17-s + 0.209·19-s − 1.61·20-s − 5.85·22-s + 23-s + 25-s − 8.30·26-s + 1.61·28-s − 2.84·29-s + 9.13·31-s − 7.32·32-s + 7.43·34-s − 35-s + 6.74·37-s + 0.398·38-s + 0.733·40-s + 3.97·41-s + ⋯ |
L(s) = 1 | + 1.34·2-s + 0.806·4-s − 0.447·5-s + 0.377·7-s − 0.259·8-s − 0.601·10-s − 0.929·11-s − 1.21·13-s + 0.508·14-s − 1.15·16-s + 0.949·17-s + 0.0480·19-s − 0.360·20-s − 1.24·22-s + 0.208·23-s + 0.200·25-s − 1.62·26-s + 0.305·28-s − 0.527·29-s + 1.64·31-s − 1.29·32-s + 1.27·34-s − 0.169·35-s + 1.10·37-s + 0.0646·38-s + 0.116·40-s + 0.621·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.143910492\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.143910492\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 - 1.90T + 2T^{2} \) |
| 11 | \( 1 + 3.08T + 11T^{2} \) |
| 13 | \( 1 + 4.36T + 13T^{2} \) |
| 17 | \( 1 - 3.91T + 17T^{2} \) |
| 19 | \( 1 - 0.209T + 19T^{2} \) |
| 29 | \( 1 + 2.84T + 29T^{2} \) |
| 31 | \( 1 - 9.13T + 31T^{2} \) |
| 37 | \( 1 - 6.74T + 37T^{2} \) |
| 41 | \( 1 - 3.97T + 41T^{2} \) |
| 43 | \( 1 - 9.51T + 43T^{2} \) |
| 47 | \( 1 + 7.70T + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 + 3.22T + 59T^{2} \) |
| 61 | \( 1 + 1.71T + 61T^{2} \) |
| 67 | \( 1 - 4.68T + 67T^{2} \) |
| 71 | \( 1 + 4.54T + 71T^{2} \) |
| 73 | \( 1 + 0.767T + 73T^{2} \) |
| 79 | \( 1 - 6.64T + 79T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 + 7.44T + 89T^{2} \) |
| 97 | \( 1 - 4.06T + 97T^{2} \) |
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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
−7.68472898437052491344129911904, −7.27323491645458292312613765980, −6.26036005446475902149723248643, −5.62794981287851375680842359014, −4.91704904870371709928491417397, −4.53732343061723509092701743720, −3.66978910396444992897996024627, −2.82365041532014148409962210617, −2.31242585207729911910147839369, −0.70606634863645320267851531312,
0.70606634863645320267851531312, 2.31242585207729911910147839369, 2.82365041532014148409962210617, 3.66978910396444992897996024627, 4.53732343061723509092701743720, 4.91704904870371709928491417397, 5.62794981287851375680842359014, 6.26036005446475902149723248643, 7.27323491645458292312613765980, 7.68472898437052491344129911904