L(s) = 1 | + 5-s − 2·7-s − 4·13-s + 3·17-s + 4·19-s − 3·23-s + 25-s + 31-s − 2·35-s + 2·37-s − 6·41-s − 8·43-s + 3·47-s − 3·49-s − 9·53-s + 12·59-s + 5·61-s − 4·65-s − 2·67-s + 12·71-s + 8·73-s + 79-s + 3·85-s − 6·89-s + 8·91-s + 4·95-s − 10·97-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.755·7-s − 1.10·13-s + 0.727·17-s + 0.917·19-s − 0.625·23-s + 1/5·25-s + 0.179·31-s − 0.338·35-s + 0.328·37-s − 0.937·41-s − 1.21·43-s + 0.437·47-s − 3/7·49-s − 1.23·53-s + 1.56·59-s + 0.640·61-s − 0.496·65-s − 0.244·67-s + 1.42·71-s + 0.936·73-s + 0.112·79-s + 0.325·85-s − 0.635·89-s + 0.838·91-s + 0.410·95-s − 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−14.08954009304021, −13.70300870988719, −13.16392696722208, −12.63334823874831, −12.23004792989189, −11.78040007415830, −11.24471090129794, −10.55614119765413, −9.920154675808086, −9.721706843310241, −9.473212612322117, −8.564021373637744, −8.096562120290151, −7.580235816265983, −6.849577968725702, −6.657333538446610, −5.879489527153846, −5.323026136018532, −4.995730222862685, −4.184192804705865, −3.467136519084027, −3.042820882305088, −2.361036494128416, −1.706990734198260, −0.8486454091722740, 0,
0.8486454091722740, 1.706990734198260, 2.361036494128416, 3.042820882305088, 3.467136519084027, 4.184192804705865, 4.995730222862685, 5.323026136018532, 5.879489527153846, 6.657333538446610, 6.849577968725702, 7.580235816265983, 8.096562120290151, 8.564021373637744, 9.473212612322117, 9.721706843310241, 9.920154675808086, 10.55614119765413, 11.24471090129794, 11.78040007415830, 12.23004792989189, 12.63334823874831, 13.16392696722208, 13.70300870988719, 14.08954009304021