L(s) = 1 | + 3-s − 5-s − 2.60·7-s + 9-s + 0.988·11-s − 3.16·13-s − 15-s + 3.38·17-s + 3.78·19-s − 2.60·21-s − 3·23-s + 25-s + 27-s + 1.60·29-s − 1.55·31-s + 0.988·33-s + 2.60·35-s − 2.20·37-s − 3.16·39-s − 6.78·41-s − 5.39·43-s − 45-s + 12.8·47-s − 0.216·49-s + 3.38·51-s + 0.551·53-s − 0.988·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.984·7-s + 0.333·9-s + 0.298·11-s − 0.878·13-s − 0.258·15-s + 0.821·17-s + 0.868·19-s − 0.568·21-s − 0.625·23-s + 0.200·25-s + 0.192·27-s + 0.297·29-s − 0.278·31-s + 0.172·33-s + 0.440·35-s − 0.363·37-s − 0.507·39-s − 1.05·41-s − 0.823·43-s − 0.149·45-s + 1.86·47-s − 0.0308·49-s + 0.474·51-s + 0.0757·53-s − 0.133·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + T \) |
good | 7 | \( 1 + 2.60T + 7T^{2} \) |
| 11 | \( 1 - 0.988T + 11T^{2} \) |
| 13 | \( 1 + 3.16T + 13T^{2} \) |
| 17 | \( 1 - 3.38T + 17T^{2} \) |
| 19 | \( 1 - 3.78T + 19T^{2} \) |
| 23 | \( 1 + 3T + 23T^{2} \) |
| 29 | \( 1 - 1.60T + 29T^{2} \) |
| 31 | \( 1 + 1.55T + 31T^{2} \) |
| 37 | \( 1 + 2.20T + 37T^{2} \) |
| 41 | \( 1 + 6.78T + 41T^{2} \) |
| 43 | \( 1 + 5.39T + 43T^{2} \) |
| 47 | \( 1 - 12.8T + 47T^{2} \) |
| 53 | \( 1 - 0.551T + 53T^{2} \) |
| 59 | \( 1 + 15.2T + 59T^{2} \) |
| 61 | \( 1 + 4.21T + 61T^{2} \) |
| 71 | \( 1 + 15.1T + 71T^{2} \) |
| 73 | \( 1 - 9.36T + 73T^{2} \) |
| 79 | \( 1 + 6.16T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 + 18.8T + 97T^{2} \) |
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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
−7.995499959009767524223874562177, −7.39672033171842019894059196322, −6.79097364738553548314163805988, −5.89452002034208907817143738383, −5.03439693115723161104719959927, −4.08217434343408751147064650316, −3.32128416399293775189043240186, −2.73069558246263557573783461711, −1.44622823566160394809383303193, 0,
1.44622823566160394809383303193, 2.73069558246263557573783461711, 3.32128416399293775189043240186, 4.08217434343408751147064650316, 5.03439693115723161104719959927, 5.89452002034208907817143738383, 6.79097364738553548314163805988, 7.39672033171842019894059196322, 7.995499959009767524223874562177