L(s) = 1 | + 2·2-s + 2·4-s − 5-s − 2·7-s − 2·10-s − 11-s − 13-s − 4·14-s − 4·16-s − 3·17-s − 19-s − 2·20-s − 2·22-s + 6·23-s + 25-s − 2·26-s − 4·28-s + 2·31-s − 8·32-s − 6·34-s + 2·35-s + 8·37-s − 2·38-s − 12·41-s + 4·43-s − 2·44-s + 12·46-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 4-s − 0.447·5-s − 0.755·7-s − 0.632·10-s − 0.301·11-s − 0.277·13-s − 1.06·14-s − 16-s − 0.727·17-s − 0.229·19-s − 0.447·20-s − 0.426·22-s + 1.25·23-s + 1/5·25-s − 0.392·26-s − 0.755·28-s + 0.359·31-s − 1.41·32-s − 1.02·34-s + 0.338·35-s + 1.31·37-s − 0.324·38-s − 1.87·41-s + 0.609·43-s − 0.301·44-s + 1.76·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.773951661\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.773951661\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−7.33804107636764490511982328014, −6.86143989321507943508410131898, −6.24834110773255394798933502398, −5.52647686000205783864583925265, −4.83115953399504002472539408109, −4.26919563076850109009598640793, −3.52432830626933526340753638942, −2.88317275040021061934728501242, −2.21021405755551016296294909446, −0.60928525862112690446159797749,
0.60928525862112690446159797749, 2.21021405755551016296294909446, 2.88317275040021061934728501242, 3.52432830626933526340753638942, 4.26919563076850109009598640793, 4.83115953399504002472539408109, 5.52647686000205783864583925265, 6.24834110773255394798933502398, 6.86143989321507943508410131898, 7.33804107636764490511982328014