| L(s) = 1 | + 2-s − 3-s + 4-s + 2·5-s − 6-s + 8-s + 9-s + 2·10-s + 11-s − 12-s + 4·13-s − 2·15-s + 16-s + 6·17-s + 18-s + 2·19-s + 2·20-s + 22-s − 24-s − 25-s + 4·26-s − 27-s − 6·29-s − 2·30-s + 2·31-s + 32-s − 33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.301·11-s − 0.288·12-s + 1.10·13-s − 0.516·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.458·19-s + 0.447·20-s + 0.213·22-s − 0.204·24-s − 1/5·25-s + 0.784·26-s − 0.192·27-s − 1.11·29-s − 0.365·30-s + 0.359·31-s + 0.176·32-s − 0.174·33-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
| \(L(1)\) |
\(\approx\) |
\(3.338519674\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.338519674\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 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 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−8.613115402786400438732963116504, −7.78162457922942700999094266600, −6.88214069039893473496004855275, −6.19024639352251723812963126128, −5.59345763706298924748190593428, −5.09475301028297681923586270591, −3.90290582311471999504150598181, −3.27984230463274719507007333781, −1.96897774543671936117435436537, −1.11644197828960166408551651917,
1.11644197828960166408551651917, 1.96897774543671936117435436537, 3.27984230463274719507007333781, 3.90290582311471999504150598181, 5.09475301028297681923586270591, 5.59345763706298924748190593428, 6.19024639352251723812963126128, 6.88214069039893473496004855275, 7.78162457922942700999094266600, 8.613115402786400438732963116504
