L(s) = 1 | − 2·2-s + 2·4-s − 5-s + 2·10-s − 5·11-s + 4·13-s − 4·16-s + 4·17-s − 5·19-s − 2·20-s + 10·22-s − 6·23-s + 25-s − 8·26-s + 5·29-s − 9·31-s + 8·32-s − 8·34-s − 10·37-s + 10·38-s − 7·41-s − 2·43-s − 10·44-s + 12·46-s − 2·47-s − 7·49-s − 2·50-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 4-s − 0.447·5-s + 0.632·10-s − 1.50·11-s + 1.10·13-s − 16-s + 0.970·17-s − 1.14·19-s − 0.447·20-s + 2.13·22-s − 1.25·23-s + 1/5·25-s − 1.56·26-s + 0.928·29-s − 1.61·31-s + 1.41·32-s − 1.37·34-s − 1.64·37-s + 1.62·38-s − 1.09·41-s − 0.304·43-s − 1.50·44-s + 1.76·46-s − 0.291·47-s − 49-s − 0.282·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 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
−10.51459133447040412716428258554, −10.04724867665766205885559716079, −8.740838314696191226965539549612, −8.203253537981750073119106447227, −7.50006448856980317673435327311, −6.31354273414722554277544867988, −4.99212686725716379469941020875, −3.50061572991346427363382995646, −1.84657171814318698813806363337, 0,
1.84657171814318698813806363337, 3.50061572991346427363382995646, 4.99212686725716379469941020875, 6.31354273414722554277544867988, 7.50006448856980317673435327311, 8.203253537981750073119106447227, 8.740838314696191226965539549612, 10.04724867665766205885559716079, 10.51459133447040412716428258554