L(s) = 1 | + (−1.41 + i)3-s + (1.73 + 1.41i)5-s + (1.00 − 2.82i)9-s − 4.89·11-s + 4.89i·13-s + (−3.86 − 0.267i)15-s − 3.46·17-s − 3.46i·19-s + 6i·23-s + (0.999 + 4.89i)25-s + (1.41 + 5.00i)27-s − 2.82i·29-s − 3.46i·31-s + (6.92 − 4.89i)33-s − 4.89i·37-s + ⋯ |
L(s) = 1 | + (−0.816 + 0.577i)3-s + (0.774 + 0.632i)5-s + (0.333 − 0.942i)9-s − 1.47·11-s + 1.35i·13-s + (−0.997 − 0.0691i)15-s − 0.840·17-s − 0.794i·19-s + 1.25i·23-s + (0.199 + 0.979i)25-s + (0.272 + 0.962i)27-s − 0.525i·29-s − 0.622i·31-s + (1.20 − 0.852i)33-s − 0.805i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0691i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0691i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0194690 + 0.562143i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0194690 + 0.562143i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.41 - i)T \) |
| 5 | \( 1 + (-1.73 - 1.41i)T \) |
good | 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 - 4.89iT - 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 3.46iT - 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 2.82iT - 29T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 + 4.89iT - 37T^{2} \) |
| 41 | \( 1 - 5.65iT - 41T^{2} \) |
| 43 | \( 1 + 8.48T + 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 + 4.89T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 8.48T + 67T^{2} \) |
| 71 | \( 1 + 9.79T + 71T^{2} \) |
| 73 | \( 1 + 9.79iT - 73T^{2} \) |
| 79 | \( 1 - 10.3iT - 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + 5.65iT - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−10.48595439242893021257717076211, −9.611437702656490931183640381005, −9.204026933881301741122306409643, −7.78535449643369668789508643213, −6.80374102685326044160797221198, −6.16097913825152331781110159189, −5.21434854427155132295743876759, −4.46499697175697823964427126900, −3.13147726395814021045594040500, −1.92873799462557679734214054048,
0.27104640821909668981851516026, 1.76180630575929780086238816138, 2.89179942929753081499958264419, 4.74600895470813989885629944487, 5.27848803366599397975705389274, 6.05270115514242760616661359686, 6.94606237783544150473105914949, 8.101028345817626227103512028151, 8.476402087672091340513630522676, 9.977131871399438359227231272112