L(s) = 1 | + (−1.58 + 0.707i)3-s − 5-s + 1.33i·7-s + (1.99 − 2.23i)9-s + 2.74·11-s − 4.49·13-s + (1.58 − 0.707i)15-s − 3.35·17-s − 3.68i·19-s + (−0.946 − 2.11i)21-s + (−0.364 + 4.78i)23-s + 25-s + (−1.57 + 4.95i)27-s − 3.60i·29-s − 0.848·31-s + ⋯ |
L(s) = 1 | + (−0.912 + 0.408i)3-s − 0.447·5-s + 0.505i·7-s + (0.666 − 0.745i)9-s + 0.829·11-s − 1.24·13-s + (0.408 − 0.182i)15-s − 0.813·17-s − 0.844i·19-s + (−0.206 − 0.461i)21-s + (−0.0759 + 0.997i)23-s + 0.200·25-s + (−0.303 + 0.952i)27-s − 0.668i·29-s − 0.152·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.338 + 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.338 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6366801916\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6366801916\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.58 - 0.707i)T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + (0.364 - 4.78i)T \) |
good | 7 | \( 1 - 1.33iT - 7T^{2} \) |
| 11 | \( 1 - 2.74T + 11T^{2} \) |
| 13 | \( 1 + 4.49T + 13T^{2} \) |
| 17 | \( 1 + 3.35T + 17T^{2} \) |
| 19 | \( 1 + 3.68iT - 19T^{2} \) |
| 29 | \( 1 + 3.60iT - 29T^{2} \) |
| 31 | \( 1 + 0.848T + 31T^{2} \) |
| 37 | \( 1 + 2.13iT - 37T^{2} \) |
| 41 | \( 1 - 5.16iT - 41T^{2} \) |
| 43 | \( 1 + 12.2iT - 43T^{2} \) |
| 47 | \( 1 + 0.193iT - 47T^{2} \) |
| 53 | \( 1 - 5.60T + 53T^{2} \) |
| 59 | \( 1 + 6.74iT - 59T^{2} \) |
| 61 | \( 1 + 1.26iT - 61T^{2} \) |
| 67 | \( 1 + 13.1iT - 67T^{2} \) |
| 71 | \( 1 + 7.83iT - 71T^{2} \) |
| 73 | \( 1 + 6.09T + 73T^{2} \) |
| 79 | \( 1 + 16.9iT - 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 + 3.98T + 89T^{2} \) |
| 97 | \( 1 + 13.7iT - 97T^{2} \) |
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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
−9.355108306948018561056446894401, −8.937999073372255668011171100008, −7.60251917552413558906499720931, −6.92327853228318722919773675196, −6.09691009391818796072107950182, −5.13544460638848389432112757561, −4.46538220881000425127057050288, −3.49421721241098465104520051443, −2.08334480845744193645999644296, −0.34096463220662690918550085809,
1.09458914803266047297969002699, 2.44734148617551927432797338459, 4.00310649951896261727768610576, 4.61478138718924174537716658686, 5.62520171775111323679241636913, 6.64909250599621302929426685844, 7.11335505772994858072382670958, 7.940116557960147009920648643274, 8.903826835748969103452267070301, 9.956131214553874801783752948585