L(s) = 1 | + 5-s − 4·7-s − 2·11-s + 6·13-s + 2·17-s + 19-s − 6·23-s + 25-s − 8·29-s − 8·31-s − 4·35-s + 10·37-s + 4·41-s + 4·43-s − 6·47-s + 9·49-s − 12·53-s − 2·55-s + 8·59-s + 2·61-s + 6·65-s + 4·67-s − 12·71-s − 10·73-s + 8·77-s − 8·79-s − 2·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 1.51·7-s − 0.603·11-s + 1.66·13-s + 0.485·17-s + 0.229·19-s − 1.25·23-s + 1/5·25-s − 1.48·29-s − 1.43·31-s − 0.676·35-s + 1.64·37-s + 0.624·41-s + 0.609·43-s − 0.875·47-s + 9/7·49-s − 1.64·53-s − 0.269·55-s + 1.04·59-s + 0.256·61-s + 0.744·65-s + 0.488·67-s − 1.42·71-s − 1.17·73-s + 0.911·77-s − 0.900·79-s − 0.219·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.195154278027595287101589658656, −7.51763594068408576761934425881, −6.57264711987684388402294646931, −5.89240349375805042329343407251, −5.58682683235758880385578107278, −4.10107534324134400504475256340, −3.50631479190343971424499020931, −2.65176951800295550388157692815, −1.45537436864446162809269751533, 0,
1.45537436864446162809269751533, 2.65176951800295550388157692815, 3.50631479190343971424499020931, 4.10107534324134400504475256340, 5.58682683235758880385578107278, 5.89240349375805042329343407251, 6.57264711987684388402294646931, 7.51763594068408576761934425881, 8.195154278027595287101589658656