L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s − 11-s − 12-s − 2·13-s + 16-s + 18-s − 2·19-s − 22-s − 24-s − 5·25-s − 2·26-s − 27-s − 6·29-s − 2·31-s + 32-s + 33-s + 36-s + 2·37-s − 2·38-s + 2·39-s − 4·43-s − 44-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s − 0.554·13-s + 1/4·16-s + 0.235·18-s − 0.458·19-s − 0.213·22-s − 0.204·24-s − 25-s − 0.392·26-s − 0.192·27-s − 1.11·29-s − 0.359·31-s + 0.176·32-s + 0.174·33-s + 1/6·36-s + 0.328·37-s − 0.324·38-s + 0.320·39-s − 0.609·43-s − 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−8.019259750876931777400838091871, −7.43580396718635232825666196862, −6.64764709700293617110788735405, −5.86088051722843540343111085134, −5.28504090410954752716150203642, −4.44329737475499316523087610458, −3.71076198654888108453853603433, −2.60622211031772570676167298329, −1.64204739427050866177734888707, 0,
1.64204739427050866177734888707, 2.60622211031772570676167298329, 3.71076198654888108453853603433, 4.44329737475499316523087610458, 5.28504090410954752716150203642, 5.86088051722843540343111085134, 6.64764709700293617110788735405, 7.43580396718635232825666196862, 8.019259750876931777400838091871