L(s) = 1 | − 3-s − 7-s − 2·9-s + 6·11-s − 5·13-s + 3·17-s − 19-s + 21-s + 3·23-s − 5·25-s + 5·27-s − 9·29-s − 4·31-s − 6·33-s − 2·37-s + 5·39-s − 8·43-s − 6·49-s − 3·51-s + 3·53-s + 57-s − 9·59-s + 10·61-s + 2·63-s − 5·67-s − 3·69-s − 6·71-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.377·7-s − 2/3·9-s + 1.80·11-s − 1.38·13-s + 0.727·17-s − 0.229·19-s + 0.218·21-s + 0.625·23-s − 25-s + 0.962·27-s − 1.67·29-s − 0.718·31-s − 1.04·33-s − 0.328·37-s + 0.800·39-s − 1.21·43-s − 6/7·49-s − 0.420·51-s + 0.412·53-s + 0.132·57-s − 1.17·59-s + 1.28·61-s + 0.251·63-s − 0.610·67-s − 0.361·69-s − 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−9.430534638052368973397733045259, −8.666881218945426773456780708719, −7.49822591496797438401690336538, −6.79687210361750407728091466084, −5.93366234401884381484558908154, −5.20686401818851827647471141246, −4.07138629192131545636061302538, −3.12424893607844884324162519175, −1.68171654176783246558951167350, 0,
1.68171654176783246558951167350, 3.12424893607844884324162519175, 4.07138629192131545636061302538, 5.20686401818851827647471141246, 5.93366234401884381484558908154, 6.79687210361750407728091466084, 7.49822591496797438401690336538, 8.666881218945426773456780708719, 9.430534638052368973397733045259