L(s) = 1 | + 2-s + 3-s − 4-s + 6-s − 2·7-s − 3·8-s + 9-s + 4·11-s − 12-s − 6·13-s − 2·14-s − 16-s + 4·17-s + 18-s + 2·19-s − 2·21-s + 4·22-s − 23-s − 3·24-s − 5·25-s − 6·26-s + 27-s + 2·28-s + 2·29-s + 4·31-s + 5·32-s + 4·33-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s − 0.755·7-s − 1.06·8-s + 1/3·9-s + 1.20·11-s − 0.288·12-s − 1.66·13-s − 0.534·14-s − 1/4·16-s + 0.970·17-s + 0.235·18-s + 0.458·19-s − 0.436·21-s + 0.852·22-s − 0.208·23-s − 0.612·24-s − 25-s − 1.17·26-s + 0.192·27-s + 0.377·28-s + 0.371·29-s + 0.718·31-s + 0.883·32-s + 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.202931933\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.202931933\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.41680599552217349849782933643, −13.92047047776520313422272180496, −12.58845419020102955872708497981, −11.98725025921574774167491193901, −9.817741003912281760084528883902, −9.300001188284360323830014666749, −7.66409820196660344666854213884, −6.11842053307024827127178733463, −4.50231733774538309156669593039, −3.14498393746938665813219257318,
3.14498393746938665813219257318, 4.50231733774538309156669593039, 6.11842053307024827127178733463, 7.66409820196660344666854213884, 9.300001188284360323830014666749, 9.817741003912281760084528883902, 11.98725025921574774167491193901, 12.58845419020102955872708497981, 13.92047047776520313422272180496, 14.41680599552217349849782933643