L(s) = 1 | + 3-s − 2·4-s + 5-s + 2·7-s + 9-s − 2·12-s − 13-s + 15-s + 4·16-s − 3·17-s − 4·19-s − 2·20-s + 2·21-s − 8·23-s − 4·25-s + 27-s − 4·28-s + 5·29-s − 31-s + 2·35-s − 2·36-s + 37-s − 39-s − 6·41-s − 8·43-s + 45-s + 3·47-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 4-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.577·12-s − 0.277·13-s + 0.258·15-s + 16-s − 0.727·17-s − 0.917·19-s − 0.447·20-s + 0.436·21-s − 1.66·23-s − 4/5·25-s + 0.192·27-s − 0.755·28-s + 0.928·29-s − 0.179·31-s + 0.338·35-s − 1/3·36-s + 0.164·37-s − 0.160·39-s − 0.937·41-s − 1.21·43-s + 0.149·45-s + 0.437·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−8.289812019619759109423310079486, −7.62363466505198381248563064317, −6.55774121903757458544896035936, −5.80653174799998452773628812102, −4.86520020905111913093774715807, −4.34415681869639972823357680004, −3.55991169571847410774924610306, −2.33128840435454651393386049365, −1.59002493368593219071271688421, 0,
1.59002493368593219071271688421, 2.33128840435454651393386049365, 3.55991169571847410774924610306, 4.34415681869639972823357680004, 4.86520020905111913093774715807, 5.80653174799998452773628812102, 6.55774121903757458544896035936, 7.62363466505198381248563064317, 8.289812019619759109423310079486