L(s) = 1 | − 7-s − 3·11-s + 13-s − 3·17-s + 2·19-s − 6·23-s + 9·29-s + 8·31-s + 10·37-s − 2·43-s − 3·47-s + 49-s − 12·59-s + 8·61-s − 8·67-s − 14·73-s + 3·77-s + 5·79-s − 12·83-s − 12·89-s − 91-s − 17·97-s + 6·101-s + 7·103-s − 6·107-s − 19·109-s − 6·113-s + ⋯ |
L(s) = 1 | − 0.377·7-s − 0.904·11-s + 0.277·13-s − 0.727·17-s + 0.458·19-s − 1.25·23-s + 1.67·29-s + 1.43·31-s + 1.64·37-s − 0.304·43-s − 0.437·47-s + 1/7·49-s − 1.56·59-s + 1.02·61-s − 0.977·67-s − 1.63·73-s + 0.341·77-s + 0.562·79-s − 1.31·83-s − 1.27·89-s − 0.104·91-s − 1.72·97-s + 0.597·101-s + 0.689·103-s − 0.580·107-s − 1.81·109-s − 0.564·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−7.891678371286758000699024997955, −6.88720330272128702619328939678, −6.27920465813494713021586226778, −5.66088244633643422777494420114, −4.67245307291987195517341252300, −4.18135032678068619831468299424, −2.98636720155307028037709109085, −2.53418839892033630277267330360, −1.25627322290419260334521146937, 0,
1.25627322290419260334521146937, 2.53418839892033630277267330360, 2.98636720155307028037709109085, 4.18135032678068619831468299424, 4.67245307291987195517341252300, 5.66088244633643422777494420114, 6.27920465813494713021586226778, 6.88720330272128702619328939678, 7.891678371286758000699024997955