L(s) = 1 | + 2-s + 4-s + 2·5-s − 7-s + 8-s − 3·9-s + 2·10-s + 4·11-s + 6·13-s − 14-s + 16-s + 6·17-s − 3·18-s + 4·19-s + 2·20-s + 4·22-s + 23-s − 25-s + 6·26-s − 28-s + 29-s − 4·31-s + 32-s + 6·34-s − 2·35-s − 3·36-s + 6·37-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.377·7-s + 0.353·8-s − 9-s + 0.632·10-s + 1.20·11-s + 1.66·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.707·18-s + 0.917·19-s + 0.447·20-s + 0.852·22-s + 0.208·23-s − 1/5·25-s + 1.17·26-s − 0.188·28-s + 0.185·29-s − 0.718·31-s + 0.176·32-s + 1.02·34-s − 0.338·35-s − 1/2·36-s + 0.986·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.776507157\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.776507157\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−7.63357673471584573583441635158, −6.72628756524947609636023043167, −6.11918140394573527266107013041, −5.75737890133230452966716051335, −5.20325619988796522167760757317, −4.01843650285911141962507515374, −3.43626689297394281494922531885, −2.89166895637896070952109710509, −1.69661777647284499644987040125, −1.03507937821618227692113510608,
1.03507937821618227692113510608, 1.69661777647284499644987040125, 2.89166895637896070952109710509, 3.43626689297394281494922531885, 4.01843650285911141962507515374, 5.20325619988796522167760757317, 5.75737890133230452966716051335, 6.11918140394573527266107013041, 6.72628756524947609636023043167, 7.63357673471584573583441635158