L(s) = 1 | − 2-s − 3-s − 4-s − 4·5-s + 6-s + 3·8-s + 9-s + 4·10-s − 5·11-s + 12-s + 13-s + 4·15-s − 16-s − 3·17-s − 18-s + 5·19-s + 4·20-s + 5·22-s + 6·23-s − 3·24-s + 11·25-s − 26-s − 27-s + 7·29-s − 4·30-s − 5·32-s + 5·33-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s − 1.78·5-s + 0.408·6-s + 1.06·8-s + 1/3·9-s + 1.26·10-s − 1.50·11-s + 0.288·12-s + 0.277·13-s + 1.03·15-s − 1/4·16-s − 0.727·17-s − 0.235·18-s + 1.14·19-s + 0.894·20-s + 1.06·22-s + 1.25·23-s − 0.612·24-s + 11/5·25-s − 0.196·26-s − 0.192·27-s + 1.29·29-s − 0.730·30-s − 0.883·32-s + 0.870·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 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
−8.457731909587718961468800495258, −8.266920508952982630633181355989, −7.35160145596218925763723286325, −6.86932913412309880980125568767, −5.24486627821193594394905030487, −4.83790678324622873376882276535, −3.90950376512155422395053188379, −2.90966914929654157418365426017, −0.970203870989400762218481231254, 0,
0.970203870989400762218481231254, 2.90966914929654157418365426017, 3.90950376512155422395053188379, 4.83790678324622873376882276535, 5.24486627821193594394905030487, 6.86932913412309880980125568767, 7.35160145596218925763723286325, 8.266920508952982630633181355989, 8.457731909587718961468800495258