L(s) = 1 | + 2-s − 3-s + 4-s + 4·5-s − 6-s + 8-s + 9-s + 4·10-s − 4·11-s − 12-s − 4·15-s + 16-s − 2·17-s + 18-s + 4·19-s + 4·20-s − 4·22-s + 4·23-s − 24-s + 11·25-s − 27-s + 4·29-s − 4·30-s − 10·31-s + 32-s + 4·33-s − 2·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.78·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 1.26·10-s − 1.20·11-s − 0.288·12-s − 1.03·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.917·19-s + 0.894·20-s − 0.852·22-s + 0.834·23-s − 0.204·24-s + 11/5·25-s − 0.192·27-s + 0.742·29-s − 0.730·30-s − 1.79·31-s + 0.176·32-s + 0.696·33-s − 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.120961917\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.120961917\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 59 | \( 1 + T \) |
good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−11.43726476834825454250783012778, −10.53041037597259470930252813861, −9.942667351479858782931283737036, −8.849632825962545132410360826727, −7.30669132252142915283690454069, −6.38763626215404051470621021393, −5.41020364394950921429932148004, −4.99033301664069494832631447632, −3.03593640856987529410776772017, −1.78043461687535210547364060412,
1.78043461687535210547364060412, 3.03593640856987529410776772017, 4.99033301664069494832631447632, 5.41020364394950921429932148004, 6.38763626215404051470621021393, 7.30669132252142915283690454069, 8.849632825962545132410360826727, 9.942667351479858782931283737036, 10.53041037597259470930252813861, 11.43726476834825454250783012778