| L(s) = 1 | + 3·3-s − 5-s + 6·9-s − 5·11-s + 5·13-s − 3·15-s + 7·17-s + 2·19-s − 2·23-s + 25-s + 9·27-s + 7·29-s − 4·31-s − 15·33-s − 6·37-s + 15·39-s + 12·41-s − 2·43-s − 6·45-s − 47-s + 21·51-s + 5·55-s + 6·57-s + 4·59-s − 4·61-s − 5·65-s + 8·67-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.447·5-s + 2·9-s − 1.50·11-s + 1.38·13-s − 0.774·15-s + 1.69·17-s + 0.458·19-s − 0.417·23-s + 1/5·25-s + 1.73·27-s + 1.29·29-s − 0.718·31-s − 2.61·33-s − 0.986·37-s + 2.40·39-s + 1.87·41-s − 0.304·43-s − 0.894·45-s − 0.145·47-s + 2.94·51-s + 0.674·55-s + 0.794·57-s + 0.520·59-s − 0.512·61-s − 0.620·65-s + 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.197509264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.197509264\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−8.937999621329745000322771267530, −8.332265077163568720704999874857, −7.79995782949092071026536073408, −7.29927337271737671794138924216, −5.99774302994776081413912070293, −5.01872075614464181579565790111, −3.84714819358078169724331248538, −3.28269243130114732472876126998, −2.50550432047388901594073309744, −1.21676181365200032832887757200,
1.21676181365200032832887757200, 2.50550432047388901594073309744, 3.28269243130114732472876126998, 3.84714819358078169724331248538, 5.01872075614464181579565790111, 5.99774302994776081413912070293, 7.29927337271737671794138924216, 7.79995782949092071026536073408, 8.332265077163568720704999874857, 8.937999621329745000322771267530
