| L(s) = 1 | − 3-s + 2·5-s − 4·7-s + 9-s − 4·11-s − 2·13-s − 2·15-s − 2·17-s + 4·21-s + 23-s − 25-s − 27-s − 2·29-s + 4·33-s − 8·35-s − 10·37-s + 2·39-s − 6·41-s + 8·43-s + 2·45-s − 8·47-s + 9·49-s + 2·51-s − 6·53-s − 8·55-s − 4·59-s + 14·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 1.51·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.516·15-s − 0.485·17-s + 0.872·21-s + 0.208·23-s − 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.696·33-s − 1.35·35-s − 1.64·37-s + 0.320·39-s − 0.937·41-s + 1.21·43-s + 0.298·45-s − 1.16·47-s + 9/7·49-s + 0.280·51-s − 0.824·53-s − 1.07·55-s − 0.520·59-s + 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−10.10716168109517185310962036964, −9.818959207850764905878975695396, −8.771125513456330198671657732746, −7.40788935293622814070976954196, −6.56638744062099278143430168894, −5.76674544434821241100411312636, −4.93114717119026292769300820712, −3.37399471176038193635666319133, −2.20156915657461643434347375119, 0,
2.20156915657461643434347375119, 3.37399471176038193635666319133, 4.93114717119026292769300820712, 5.76674544434821241100411312636, 6.56638744062099278143430168894, 7.40788935293622814070976954196, 8.771125513456330198671657732746, 9.818959207850764905878975695396, 10.10716168109517185310962036964
