L(s) = 1 | − 2-s − 3-s − 4-s + 6-s + 3·8-s − 2·9-s + 12-s + 2·13-s − 16-s + 2·17-s + 2·18-s − 6·19-s + 3·23-s − 3·24-s − 2·26-s + 5·27-s + 7·29-s − 2·31-s − 5·32-s − 2·34-s + 2·36-s + 8·37-s + 6·38-s − 2·39-s − 5·41-s − 7·43-s − 3·46-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.408·6-s + 1.06·8-s − 2/3·9-s + 0.288·12-s + 0.554·13-s − 1/4·16-s + 0.485·17-s + 0.471·18-s − 1.37·19-s + 0.625·23-s − 0.612·24-s − 0.392·26-s + 0.962·27-s + 1.29·29-s − 0.359·31-s − 0.883·32-s − 0.342·34-s + 1/3·36-s + 1.31·37-s + 0.973·38-s − 0.320·39-s − 0.780·41-s − 1.06·43-s − 0.442·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−9.238253215525544755854749779113, −8.497657177017218580507120470705, −8.012788101091726547559672016182, −6.80293996644022726251615709133, −6.02010512042412109072101360575, −5.04721464747136835456471164127, −4.24828178185965173155123273711, −2.95878100685676755087850454730, −1.35247945969232237010313996352, 0,
1.35247945969232237010313996352, 2.95878100685676755087850454730, 4.24828178185965173155123273711, 5.04721464747136835456471164127, 6.02010512042412109072101360575, 6.80293996644022726251615709133, 8.012788101091726547559672016182, 8.497657177017218580507120470705, 9.238253215525544755854749779113