L(s) = 1 | + 2.41·3-s − 5-s + 2.82·9-s − 4.82·11-s − 0.828·13-s − 2.41·15-s + 0.828·17-s − 2.82·19-s + 2.41·23-s + 25-s − 0.414·27-s − 29-s − 6·31-s − 11.6·33-s − 1.99·39-s + 2.17·41-s − 6.41·43-s − 2.82·45-s + 2·47-s + 1.99·51-s − 6.82·53-s + 4.82·55-s − 6.82·57-s − 12.4·59-s + 11.4·61-s + 0.828·65-s − 12.4·67-s + ⋯ |
L(s) = 1 | + 1.39·3-s − 0.447·5-s + 0.942·9-s − 1.45·11-s − 0.229·13-s − 0.623·15-s + 0.200·17-s − 0.648·19-s + 0.503·23-s + 0.200·25-s − 0.0797·27-s − 0.185·29-s − 1.07·31-s − 2.02·33-s − 0.320·39-s + 0.339·41-s − 0.978·43-s − 0.421·45-s + 0.291·47-s + 0.280·51-s − 0.937·53-s + 0.651·55-s − 0.904·57-s − 1.62·59-s + 1.47·61-s + 0.102·65-s − 1.51·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 2.41T + 3T^{2} \) |
| 11 | \( 1 + 4.82T + 11T^{2} \) |
| 13 | \( 1 + 0.828T + 13T^{2} \) |
| 17 | \( 1 - 0.828T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 - 2.41T + 23T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 2.17T + 41T^{2} \) |
| 43 | \( 1 + 6.41T + 43T^{2} \) |
| 47 | \( 1 - 2T + 47T^{2} \) |
| 53 | \( 1 + 6.82T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 + 4.82T + 73T^{2} \) |
| 79 | \( 1 + 9.17T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 + 2.65T + 89T^{2} \) |
| 97 | \( 1 + 0.343T + 97T^{2} \) |
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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
−8.119928995173777812942781761215, −7.58566790301385592765185910177, −6.97629223454409264378965851240, −5.80010369114703331750718807753, −4.97011694112183452563035874189, −4.11262001685906042033959309059, −3.22173237827856444670520136259, −2.66649123858515447411599559523, −1.74495896908886396167095089638, 0,
1.74495896908886396167095089638, 2.66649123858515447411599559523, 3.22173237827856444670520136259, 4.11262001685906042033959309059, 4.97011694112183452563035874189, 5.80010369114703331750718807753, 6.97629223454409264378965851240, 7.58566790301385592765185910177, 8.119928995173777812942781761215