L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 2·7-s + 8-s + 9-s − 10-s + 11-s − 12-s + 4·13-s − 2·14-s + 15-s + 16-s + 17-s + 18-s − 2·19-s − 20-s + 2·21-s + 22-s − 6·23-s − 24-s + 25-s + 4·26-s − 27-s − 2·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s − 0.288·12-s + 1.10·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.458·19-s − 0.223·20-s + 0.436·21-s + 0.213·22-s − 1.25·23-s − 0.204·24-s + 1/5·25-s + 0.784·26-s − 0.192·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.083916772\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.083916772\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−8.037206948194523174051444248641, −7.21560262404786146706788478227, −6.49468231063830879147467725881, −6.02810078354075585425584173963, −5.36956118307063847624032023226, −4.27612137197952264322258233950, −3.88056860413619564871792966020, −3.06370209869423344815394864016, −1.92502601871261076237321702604, −0.71100367724734275127993531334,
0.71100367724734275127993531334, 1.92502601871261076237321702604, 3.06370209869423344815394864016, 3.88056860413619564871792966020, 4.27612137197952264322258233950, 5.36956118307063847624032023226, 6.02810078354075585425584173963, 6.49468231063830879147467725881, 7.21560262404786146706788478227, 8.037206948194523174051444248641