L(s) = 1 | + 2-s − 4-s + 2·7-s − 3·8-s − 3·9-s − 6·11-s − 2·13-s + 2·14-s − 16-s + 2·17-s − 3·18-s − 2·19-s − 6·22-s − 2·23-s − 2·26-s − 2·28-s − 29-s + 2·31-s + 5·32-s + 2·34-s + 3·36-s − 10·37-s − 2·38-s + 2·41-s − 8·43-s + 6·44-s − 2·46-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.755·7-s − 1.06·8-s − 9-s − 1.80·11-s − 0.554·13-s + 0.534·14-s − 1/4·16-s + 0.485·17-s − 0.707·18-s − 0.458·19-s − 1.27·22-s − 0.417·23-s − 0.392·26-s − 0.377·28-s − 0.185·29-s + 0.359·31-s + 0.883·32-s + 0.342·34-s + 1/2·36-s − 1.64·37-s − 0.324·38-s + 0.312·41-s − 1.21·43-s + 0.904·44-s − 0.294·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 29 | \( 1 + T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 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
−10.14265173502198703465081591225, −8.947550273198981438091464686696, −8.252858064103973279037456381237, −7.52061109178487627462618377984, −6.01134697239891133882099417885, −5.27104240153111024683298781878, −4.71404743035249738453779867980, −3.35895406827147515192265858691, −2.37337806558837572700935795284, 0,
2.37337806558837572700935795284, 3.35895406827147515192265858691, 4.71404743035249738453779867980, 5.27104240153111024683298781878, 6.01134697239891133882099417885, 7.52061109178487627462618377984, 8.252858064103973279037456381237, 8.947550273198981438091464686696, 10.14265173502198703465081591225