| L(s) = 1 | + 1.41i·2-s + 1.73i·3-s − 2.00·4-s + (−1.73 − 4.68i)5-s − 2.44·6-s − 7.48·7-s − 2.82i·8-s − 2.99·9-s + (6.63 − 2.45i)10-s + 2.31i·11-s − 3.46i·12-s + 6.23i·13-s − 10.5i·14-s + (8.12 − 3.00i)15-s + 4.00·16-s + 2.91·17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.500·4-s + (−0.346 − 0.937i)5-s − 0.408·6-s − 1.06·7-s − 0.353i·8-s − 0.333·9-s + (0.663 − 0.245i)10-s + 0.210i·11-s − 0.288i·12-s + 0.479i·13-s − 0.756i·14-s + (0.541 − 0.200i)15-s + 0.250·16-s + 0.171·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.706 - 0.708i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.706 - 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.247407847\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.247407847\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 - 1.73iT \) |
| 5 | \( 1 + (1.73 + 4.68i)T \) |
| 23 | \( 1 + (-9.64 - 20.8i)T \) |
| good | 7 | \( 1 + 7.48T + 49T^{2} \) |
| 11 | \( 1 - 2.31iT - 121T^{2} \) |
| 13 | \( 1 - 6.23iT - 169T^{2} \) |
| 17 | \( 1 - 2.91T + 289T^{2} \) |
| 19 | \( 1 + 25.2iT - 361T^{2} \) |
| 29 | \( 1 - 42.7T + 841T^{2} \) |
| 31 | \( 1 - 20.9T + 961T^{2} \) |
| 37 | \( 1 - 62.7T + 1.36e3T^{2} \) |
| 41 | \( 1 - 33.4T + 1.68e3T^{2} \) |
| 43 | \( 1 - 41.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + 38.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 31.9T + 2.80e3T^{2} \) |
| 59 | \( 1 - 22.5T + 3.48e3T^{2} \) |
| 61 | \( 1 - 17.0iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 77.7T + 4.48e3T^{2} \) |
| 71 | \( 1 + 134.T + 5.04e3T^{2} \) |
| 73 | \( 1 - 4.53iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 28.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 19.1T + 6.88e3T^{2} \) |
| 89 | \( 1 + 18.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 145.T + 9.40e3T^{2} \) |
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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
−10.00384132239717701360937372458, −9.360308811259154699988321646081, −8.810650927247391868157398332514, −7.77991299968426431408911801273, −6.80638801621335055445894930968, −5.88817693687341670570456820017, −4.83416918894956081173341450847, −4.16260951316626587450233145676, −2.91888830312984773707740953705, −0.70557494061066383511989579997,
0.791103912910045336880979585557, 2.57828362228274458875776401370, 3.18023066172854347082772159838, 4.31160518060030715574112306040, 5.95085405258317769531248697214, 6.48541966609478986592082813168, 7.63380694684405152729139929225, 8.343981691850978827844577173564, 9.565751011225310927338425331275, 10.27098744496305790232891367042
