L(s) = 1 | + 1.76·3-s + 5-s − 1.76·7-s + 0.103·9-s − 4.62·11-s + 1.76·15-s + 2.38·19-s − 3.10·21-s − 4·23-s + 25-s − 5.10·27-s − 7.25·29-s − 1.13·31-s − 8.14·33-s − 1.76·35-s + 37-s + 0.896·41-s − 9.25·43-s + 0.103·45-s + 8.74·47-s − 3.89·49-s + 8.35·53-s − 4.62·55-s + 4.20·57-s + 8.11·59-s + 5.04·61-s − 0.181·63-s + ⋯ |
L(s) = 1 | + 1.01·3-s + 0.447·5-s − 0.665·7-s + 0.0343·9-s − 1.39·11-s + 0.454·15-s + 0.547·19-s − 0.677·21-s − 0.834·23-s + 0.200·25-s − 0.982·27-s − 1.34·29-s − 0.203·31-s − 1.41·33-s − 0.297·35-s + 0.164·37-s + 0.140·41-s − 1.41·43-s + 0.0153·45-s + 1.27·47-s − 0.556·49-s + 1.14·53-s − 0.623·55-s + 0.557·57-s + 1.05·59-s + 0.646·61-s − 0.0228·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 - T \) |
good | 3 | \( 1 - 1.76T + 3T^{2} \) |
| 7 | \( 1 + 1.76T + 7T^{2} \) |
| 11 | \( 1 + 4.62T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 2.38T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 7.25T + 29T^{2} \) |
| 31 | \( 1 + 1.13T + 31T^{2} \) |
| 41 | \( 1 - 0.896T + 41T^{2} \) |
| 43 | \( 1 + 9.25T + 43T^{2} \) |
| 47 | \( 1 - 8.74T + 47T^{2} \) |
| 53 | \( 1 - 8.35T + 53T^{2} \) |
| 59 | \( 1 - 8.11T + 59T^{2} \) |
| 61 | \( 1 - 5.04T + 61T^{2} \) |
| 67 | \( 1 + 16.1T + 67T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 + 18.0T + 83T^{2} \) |
| 89 | \( 1 - 1.45T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−8.495875054585360795802095945980, −7.61704325784401605453653018508, −7.14180844127611195199551233174, −5.84757474444869880476057354559, −5.53109873355808720688838349320, −4.28305067836020154265702049463, −3.29075223151075227607716128858, −2.70432794332268591438512901533, −1.83530597956227026303118743508, 0,
1.83530597956227026303118743508, 2.70432794332268591438512901533, 3.29075223151075227607716128858, 4.28305067836020154265702049463, 5.53109873355808720688838349320, 5.84757474444869880476057354559, 7.14180844127611195199551233174, 7.61704325784401605453653018508, 8.495875054585360795802095945980