L(s) = 1 | + (−0.273 + 1.38i)2-s + (1.55 + 0.758i)3-s + (−1.85 − 0.758i)4-s + (−1.47 + 1.95i)6-s + 3.56i·7-s + (1.55 − 2.36i)8-s + (1.85 + 2.36i)9-s − 4.20·11-s + (−2.30 − 2.58i)12-s + 2.70·13-s + (−4.94 − 0.973i)14-s + (2.85 + 2.80i)16-s − 0.828i·17-s + (−3.78 + 1.92i)18-s + 5.07i·19-s + ⋯ |
L(s) = 1 | + (−0.193 + 0.981i)2-s + (0.899 + 0.437i)3-s + (−0.925 − 0.379i)4-s + (−0.603 + 0.797i)6-s + 1.34i·7-s + (0.550 − 0.834i)8-s + (0.616 + 0.787i)9-s − 1.26·11-s + (−0.666 − 0.745i)12-s + 0.749·13-s + (−1.32 − 0.260i)14-s + (0.712 + 0.701i)16-s − 0.200i·17-s + (−0.891 + 0.453i)18-s + 1.16i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.666 - 0.745i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.666 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.557987 + 1.24661i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.557987 + 1.24661i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.273 - 1.38i)T \) |
| 3 | \( 1 + (-1.55 - 0.758i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 3.56iT - 7T^{2} \) |
| 11 | \( 1 + 4.20T + 11T^{2} \) |
| 13 | \( 1 - 2.70T + 13T^{2} \) |
| 17 | \( 1 + 0.828iT - 17T^{2} \) |
| 19 | \( 1 - 5.07iT - 19T^{2} \) |
| 23 | \( 1 - 1.09T + 23T^{2} \) |
| 29 | \( 1 + 5.55iT - 29T^{2} \) |
| 31 | \( 1 + 6.59iT - 31T^{2} \) |
| 37 | \( 1 - 5.40T + 37T^{2} \) |
| 41 | \( 1 + 10.2iT - 41T^{2} \) |
| 43 | \( 1 - 0.531iT - 43T^{2} \) |
| 47 | \( 1 - 6.22T + 47T^{2} \) |
| 53 | \( 1 - 5.55iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 0.701T + 61T^{2} \) |
| 67 | \( 1 + 2.04iT - 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 7.70T + 73T^{2} \) |
| 79 | \( 1 + 7.12iT - 79T^{2} \) |
| 83 | \( 1 - 3.11T + 83T^{2} \) |
| 89 | \( 1 - 4.72iT - 89T^{2} \) |
| 97 | \( 1 + 8.10T + 97T^{2} \) |
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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
−12.36204068638758635166015882750, −10.83149545921053663709718687619, −9.846693819763601134270087611419, −9.063993060497432910770503270068, −8.223272668661541289103430197430, −7.64064624213978391399390823198, −6.03047721130242110340487955725, −5.27250910433095717077873714338, −3.93552589258612856189814586927, −2.39910027569157357029289583304,
1.08287917129573480032568601736, 2.71306679449250231284640212447, 3.72977384243359151588540675288, 4.89752674401541881697235875089, 6.85610317253940320229624868625, 7.80463919377036841263323981193, 8.573377753536203383256367974673, 9.618394477459269690648487101708, 10.55327570140271168810577759571, 11.14286366929421312642585109259