| L(s) = 1 | + 3-s − 7-s + 9-s − 1.05·11-s + 3.55·13-s − 4.49·17-s − 19-s − 21-s − 7.43·23-s − 5·25-s + 27-s − 9.55·29-s + 6.61·31-s − 1.05·33-s − 8.61·37-s + 3.55·39-s − 0.117·41-s − 1.88·43-s − 0.941·47-s + 49-s − 4.49·51-s − 2.44·53-s − 57-s + 12.9·59-s + 6.99·61-s − 63-s + 0.824·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 0.333·9-s − 0.319·11-s + 0.986·13-s − 1.09·17-s − 0.229·19-s − 0.218·21-s − 1.55·23-s − 25-s + 0.192·27-s − 1.77·29-s + 1.18·31-s − 0.184·33-s − 1.41·37-s + 0.569·39-s − 0.0183·41-s − 0.287·43-s − 0.137·47-s + 0.142·49-s − 0.629·51-s − 0.335·53-s − 0.132·57-s + 1.69·59-s + 0.895·61-s − 0.125·63-s + 0.100·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 1.05T + 11T^{2} \) |
| 13 | \( 1 - 3.55T + 13T^{2} \) |
| 17 | \( 1 + 4.49T + 17T^{2} \) |
| 23 | \( 1 + 7.43T + 23T^{2} \) |
| 29 | \( 1 + 9.55T + 29T^{2} \) |
| 31 | \( 1 - 6.61T + 31T^{2} \) |
| 37 | \( 1 + 8.61T + 37T^{2} \) |
| 41 | \( 1 + 0.117T + 41T^{2} \) |
| 43 | \( 1 + 1.88T + 43T^{2} \) |
| 47 | \( 1 + 0.941T + 47T^{2} \) |
| 53 | \( 1 + 2.44T + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 - 6.99T + 61T^{2} \) |
| 67 | \( 1 - 0.824T + 67T^{2} \) |
| 71 | \( 1 - 1.43T + 71T^{2} \) |
| 73 | \( 1 - 0.117T + 73T^{2} \) |
| 79 | \( 1 + 3.67T + 79T^{2} \) |
| 83 | \( 1 + 9.67T + 83T^{2} \) |
| 89 | \( 1 - 5.11T + 89T^{2} \) |
| 97 | \( 1 + 12.0T + 97T^{2} \) |
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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
−8.384009629925221326744658170470, −7.67378433711953986256666906695, −6.77670272815470916751001441231, −6.11561959575964186374835783784, −5.28853088257154114054014988122, −4.06068804146089013802646969735, −3.70443715222408585438457468847, −2.49179577074201826093883513892, −1.70592124437086147635292832874, 0,
1.70592124437086147635292832874, 2.49179577074201826093883513892, 3.70443715222408585438457468847, 4.06068804146089013802646969735, 5.28853088257154114054014988122, 6.11561959575964186374835783784, 6.77670272815470916751001441231, 7.67378433711953986256666906695, 8.384009629925221326744658170470
