L(s) = 1 | + 1.41·3-s − 1.41i·5-s + (−2.44 − i)7-s − 0.999·9-s + 3.46i·11-s + 4.24i·13-s − 2.00i·15-s + 4.89i·17-s + 4.24·19-s + (−3.46 − 1.41i)21-s + 2.99·25-s − 5.65·27-s + 4.89·31-s + 4.89i·33-s + (−1.41 + 3.46i)35-s + ⋯ |
L(s) = 1 | + 0.816·3-s − 0.632i·5-s + (−0.925 − 0.377i)7-s − 0.333·9-s + 1.04i·11-s + 1.17i·13-s − 0.516i·15-s + 1.18i·17-s + 0.973·19-s + (−0.755 − 0.308i)21-s + 0.599·25-s − 1.08·27-s + 0.879·31-s + 0.852i·33-s + (−0.239 + 0.585i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.377 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.377 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.612813540\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.612813540\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (2.44 + i)T \) |
good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 - 4.24iT - 13T^{2} \) |
| 17 | \( 1 - 4.89iT - 17T^{2} \) |
| 19 | \( 1 - 4.24T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 4.89T + 31T^{2} \) |
| 37 | \( 1 + 6.92T + 37T^{2} \) |
| 41 | \( 1 - 9.79iT - 41T^{2} \) |
| 43 | \( 1 - 3.46iT - 43T^{2} \) |
| 47 | \( 1 + 4.89T + 47T^{2} \) |
| 53 | \( 1 - 13.8T + 53T^{2} \) |
| 59 | \( 1 - 7.07T + 59T^{2} \) |
| 61 | \( 1 + 4.24iT - 61T^{2} \) |
| 67 | \( 1 + 10.3iT - 67T^{2} \) |
| 71 | \( 1 + 6iT - 71T^{2} \) |
| 73 | \( 1 - 4.89iT - 73T^{2} \) |
| 79 | \( 1 - 14iT - 79T^{2} \) |
| 83 | \( 1 + 9.89T + 83T^{2} \) |
| 89 | \( 1 - 14.6iT - 89T^{2} \) |
| 97 | \( 1 - 4.89iT - 97T^{2} \) |
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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
−9.515623196329399542212673382501, −8.648123735137046736639378784913, −8.039520803853653258543850856148, −7.02578011902703694816603200913, −6.43503934307512630470447667122, −5.28029665233787629966735827296, −4.31342421591769038259941766989, −3.55488630590315406052035984153, −2.52646499416516191181961122249, −1.38450297121084948914029197418,
0.54931187676115688428746789701, 2.58496958826179884167791594837, 3.03915795818782776745798006816, 3.63680198643070473208835046390, 5.32594936925029176016326144533, 5.80345140175968140173791188695, 6.91665561622795032902604863841, 7.50742207135134326678506149083, 8.662429596446078608080643742667, 8.828034021575194820559459235263