| L(s) = 1 | − 7-s − 3·9-s − 5·11-s + 6·13-s − 4·17-s − 6·19-s − 3·23-s − 3·29-s + 2·31-s − 7·37-s − 4·41-s + 7·43-s + 2·47-s + 49-s + 10·53-s − 14·59-s + 4·61-s + 3·63-s − 3·67-s − 13·71-s − 16·73-s + 5·77-s + 79-s + 9·81-s − 10·83-s + 10·89-s − 6·91-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 9-s − 1.50·11-s + 1.66·13-s − 0.970·17-s − 1.37·19-s − 0.625·23-s − 0.557·29-s + 0.359·31-s − 1.15·37-s − 0.624·41-s + 1.06·43-s + 0.291·47-s + 1/7·49-s + 1.37·53-s − 1.82·59-s + 0.512·61-s + 0.377·63-s − 0.366·67-s − 1.54·71-s − 1.87·73-s + 0.569·77-s + 0.112·79-s + 81-s − 1.09·83-s + 1.05·89-s − 0.628·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−10.30069404637224061553152573941, −8.715346505355531768469951546238, −8.653246877942416184154360916759, −7.45914406846797526842515158316, −6.22939814388948458334375419498, −5.72804536779750531779511394240, −4.43281388321550191752484101590, −3.27570571152979309341121964692, −2.17281069856537899044144271139, 0,
2.17281069856537899044144271139, 3.27570571152979309341121964692, 4.43281388321550191752484101590, 5.72804536779750531779511394240, 6.22939814388948458334375419498, 7.45914406846797526842515158316, 8.653246877942416184154360916759, 8.715346505355531768469951546238, 10.30069404637224061553152573941
