L(s) = 1 | + (1.84 − 0.277i)3-s + (−0.365 + 0.930i)5-s + (0.997 + 0.0747i)7-s + (2.35 − 0.726i)9-s + (−0.414 + 1.81i)15-s + (1.85 − 0.139i)21-s + (−0.246 − 0.167i)23-s + (−0.733 − 0.680i)25-s + (2.45 − 1.18i)27-s + (−1.48 − 0.716i)29-s + (−0.433 + 0.900i)35-s + (−1.19 − 1.49i)41-s + (−0.367 + 0.460i)43-s + (−0.184 + 2.45i)45-s + (−1.14 + 1.06i)47-s + ⋯ |
L(s) = 1 | + (1.84 − 0.277i)3-s + (−0.365 + 0.930i)5-s + (0.997 + 0.0747i)7-s + (2.35 − 0.726i)9-s + (−0.414 + 1.81i)15-s + (1.85 − 0.139i)21-s + (−0.246 − 0.167i)23-s + (−0.733 − 0.680i)25-s + (2.45 − 1.18i)27-s + (−1.48 − 0.716i)29-s + (−0.433 + 0.900i)35-s + (−1.19 − 1.49i)41-s + (−0.367 + 0.460i)43-s + (−0.184 + 2.45i)45-s + (−1.14 + 1.06i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.603551923\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.603551923\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.365 - 0.930i)T \) |
| 7 | \( 1 + (-0.997 - 0.0747i)T \) |
good | 3 | \( 1 + (-1.84 + 0.277i)T + (0.955 - 0.294i)T^{2} \) |
| 11 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 13 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 17 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.246 + 0.167i)T + (0.365 + 0.930i)T^{2} \) |
| 29 | \( 1 + (1.48 + 0.716i)T + (0.623 + 0.781i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.988 - 0.149i)T^{2} \) |
| 41 | \( 1 + (1.19 + 1.49i)T + (-0.222 + 0.974i)T^{2} \) |
| 43 | \( 1 + (0.367 - 0.460i)T + (-0.222 - 0.974i)T^{2} \) |
| 47 | \( 1 + (1.14 - 1.06i)T + (0.0747 - 0.997i)T^{2} \) |
| 53 | \( 1 + (0.988 + 0.149i)T^{2} \) |
| 59 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 61 | \( 1 + (-0.147 - 1.97i)T + (-0.988 + 0.149i)T^{2} \) |
| 67 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 73 | \( 1 + (-0.0747 - 0.997i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.443 + 1.94i)T + (-0.900 - 0.433i)T^{2} \) |
| 89 | \( 1 + (-1.57 + 0.487i)T + (0.826 - 0.563i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.588748999409221035322665954146, −7.85239515845810220773035705573, −7.49795117210360886183198001838, −6.84459582475502283175840771983, −5.80549264801520070020772142617, −4.54322942855831142027423992639, −3.86962375646484779367262247053, −3.13534936881634217172229259402, −2.30672404267607447348123258823, −1.63250811442692923365637826157,
1.50894754381764573390088109550, 2.05533025984916761279385400205, 3.35889365869220800035127539196, 3.83918562587801451205741819334, 4.78842123748015075168481342938, 5.19988926767350554356229827910, 6.69877361271441456522810492630, 7.63961765500033187997595891714, 8.029006411404705555126925863691, 8.511649867443095482591020521985