L(s) = 1 | + 3-s − 2·5-s − 2·7-s + 9-s − 2·11-s + 13-s − 2·15-s + 6·17-s + 2·19-s − 2·21-s − 25-s + 27-s − 2·29-s + 2·31-s − 2·33-s + 4·35-s − 10·37-s + 39-s + 2·41-s − 8·43-s − 2·45-s − 2·47-s − 3·49-s + 6·51-s + 2·53-s + 4·55-s + 2·57-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.894·5-s − 0.755·7-s + 1/3·9-s − 0.603·11-s + 0.277·13-s − 0.516·15-s + 1.45·17-s + 0.458·19-s − 0.436·21-s − 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.359·31-s − 0.348·33-s + 0.676·35-s − 1.64·37-s + 0.160·39-s + 0.312·41-s − 1.21·43-s − 0.298·45-s − 0.291·47-s − 3/7·49-s + 0.840·51-s + 0.274·53-s + 0.539·55-s + 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 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 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−8.472291229729577561267446968616, −7.70238653866752874546628860503, −7.32185771162981256016187886050, −6.25627621415601579990688034936, −5.39011374636217047998977702840, −4.39643067263263789920556453539, −3.34438303780590532331400946205, −3.10707452594267652556920073153, −1.55357312587466315401743793249, 0,
1.55357312587466315401743793249, 3.10707452594267652556920073153, 3.34438303780590532331400946205, 4.39643067263263789920556453539, 5.39011374636217047998977702840, 6.25627621415601579990688034936, 7.32185771162981256016187886050, 7.70238653866752874546628860503, 8.472291229729577561267446968616