| L(s) = 1 | − 2·4-s + 2.05·7-s + 0.990·13-s + 4·16-s + 2.60·19-s − 5·25-s − 4.11·28-s + 1.54·37-s + 9.98·43-s − 2.76·49-s − 1.98·52-s + 15.3·61-s − 8·64-s + 14.9·67-s − 6.90·73-s − 5.21·76-s − 13.4·79-s + 2.03·91-s + 7.15·97-s + 10·100-s − 18.8·103-s + 16.7·109-s + 8.23·112-s + ⋯ |
| L(s) = 1 | − 4-s + 0.778·7-s + 0.274·13-s + 16-s + 0.598·19-s − 25-s − 0.778·28-s + 0.253·37-s + 1.52·43-s − 0.394·49-s − 0.274·52-s + 1.97·61-s − 64-s + 1.83·67-s − 0.807·73-s − 0.598·76-s − 1.50·79-s + 0.213·91-s + 0.726·97-s + 100-s − 1.85·103-s + 1.60·109-s + 0.778·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.695696856\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.695696856\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 2.05T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 0.990T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 2.60T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 37 | \( 1 - 1.54T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 9.98T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 15.3T + 61T^{2} \) |
| 67 | \( 1 - 14.9T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 6.90T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 7.15T + 97T^{2} \) |
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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
−7.84334611376857065095749627546, −7.30187366333703676118116701113, −6.26423435778767803895445111909, −5.53019342233904508186275858795, −5.04288938511725188918206500977, −4.19374164339101012690212328007, −3.72948105537746314657414237465, −2.66582664058345964745955076156, −1.60121849748486198047545473844, −0.67430052622981148030174129339,
0.67430052622981148030174129339, 1.60121849748486198047545473844, 2.66582664058345964745955076156, 3.72948105537746314657414237465, 4.19374164339101012690212328007, 5.04288938511725188918206500977, 5.53019342233904508186275858795, 6.26423435778767803895445111909, 7.30187366333703676118116701113, 7.84334611376857065095749627546
