L(s) = 1 | + 2-s + 4-s + 7-s + 8-s − 4·11-s + 6·13-s + 14-s + 16-s + 2·17-s − 4·22-s + 6·26-s + 28-s − 6·29-s + 8·31-s + 32-s + 2·34-s + 10·37-s − 2·41-s − 4·43-s − 4·44-s + 8·47-s + 49-s + 6·52-s − 2·53-s + 56-s − 6·58-s + 8·59-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 1.20·11-s + 1.66·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.852·22-s + 1.17·26-s + 0.188·28-s − 1.11·29-s + 1.43·31-s + 0.176·32-s + 0.342·34-s + 1.64·37-s − 0.312·41-s − 0.609·43-s − 0.603·44-s + 1.16·47-s + 1/7·49-s + 0.832·52-s − 0.274·53-s + 0.133·56-s − 0.787·58-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.237547026\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.237547026\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.363608030078191123725662506219, −8.027088624732001759148453980283, −7.15015901425524002945866203830, −6.17150280411636765429661983697, −5.66533618240794239616734860431, −4.84400878985485571871578959566, −3.99067474905046324961213028566, −3.16804411819433045396819389047, −2.22797785605031286332443349075, −1.02699351351041065390998044717,
1.02699351351041065390998044717, 2.22797785605031286332443349075, 3.16804411819433045396819389047, 3.99067474905046324961213028566, 4.84400878985485571871578959566, 5.66533618240794239616734860431, 6.17150280411636765429661983697, 7.15015901425524002945866203830, 8.027088624732001759148453980283, 8.363608030078191123725662506219