| L(s) = 1 | + 2-s + 4-s − 2.64·7-s + 8-s − 3.74i·11-s + 4.24i·13-s − 2.64·14-s + 16-s − 5.29·19-s − 3.74i·22-s + (3 + 3.74i)23-s − 5·25-s + 4.24i·26-s − 2.64·28-s − 6·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.999·7-s + 0.353·8-s − 1.12i·11-s + 1.17i·13-s − 0.707·14-s + 0.250·16-s − 1.21·19-s − 0.797i·22-s + (0.625 + 0.780i)23-s − 25-s + 0.832i·26-s − 0.500·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.625 - 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.625 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.074655373\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.074655373\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + 2.64T \) |
| 23 | \( 1 + (-3 - 3.74i)T \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 3.74iT - 11T^{2} \) |
| 13 | \( 1 - 4.24iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 5.29T + 19T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 8.48iT - 31T^{2} \) |
| 37 | \( 1 - 11.2iT - 37T^{2} \) |
| 41 | \( 1 - 5.65iT - 41T^{2} \) |
| 43 | \( 1 + 11.2iT - 43T^{2} \) |
| 47 | \( 1 + 2.82iT - 47T^{2} \) |
| 53 | \( 1 + 3.74iT - 53T^{2} \) |
| 59 | \( 1 - 9.89iT - 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 - 11.2iT - 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 8.48iT - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 + 15.8T + 89T^{2} \) |
| 97 | \( 1 + 5.29T + 97T^{2} \) |
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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
−8.913487933539655488684559859833, −8.432638664209062037869342488450, −7.18632218290938902183716625536, −6.69887826432390189549001496476, −5.97348792941557624661012345547, −5.27571459017992269306066473669, −4.14769619263461526932399963377, −3.55877467082129725539108911657, −2.69630869581882641332146538740, −1.51764621664569877255707739830,
0.24390512197382370615077495304, 2.03559795432283645106603983425, 2.78038594758042144963002457247, 3.85887786935502340177450220201, 4.40035835844814674010577515631, 5.56721320969627415274843338869, 6.03126149738521130263448017495, 6.95202293576841434877086889388, 7.53815432585048605642359520524, 8.382512760294483559390147134955
