| L(s) = 1 | + (−1.46 + 1.36i)2-s + (0.265 − 3.99i)4-s + 1.87i·7-s + (5.06 + 6.19i)8-s − 15.1i·11-s − 18.1·13-s + (−2.55 − 2.73i)14-s + (−15.8 − 2.11i)16-s + 12.6·17-s + 28.1i·19-s + (20.6 + 22.0i)22-s + 10.9i·23-s + (26.4 − 24.7i)26-s + (7.46 + 0.496i)28-s + 9.95·29-s + ⋯ |
| L(s) = 1 | + (−0.730 + 0.683i)2-s + (0.0663 − 0.997i)4-s + 0.267i·7-s + (0.633 + 0.773i)8-s − 1.37i·11-s − 1.39·13-s + (−0.182 − 0.195i)14-s + (−0.991 − 0.132i)16-s + 0.746·17-s + 1.48i·19-s + (0.938 + 1.00i)22-s + 0.475i·23-s + (1.01 − 0.952i)26-s + (0.266 + 0.0177i)28-s + 0.343·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0663 - 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0663 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9909663217\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9909663217\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.46 - 1.36i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 1.87iT - 49T^{2} \) |
| 11 | \( 1 + 15.1iT - 121T^{2} \) |
| 13 | \( 1 + 18.1T + 169T^{2} \) |
| 17 | \( 1 - 12.6T + 289T^{2} \) |
| 19 | \( 1 - 28.1iT - 361T^{2} \) |
| 23 | \( 1 - 10.9iT - 529T^{2} \) |
| 29 | \( 1 - 9.95T + 841T^{2} \) |
| 31 | \( 1 - 24.4iT - 961T^{2} \) |
| 37 | \( 1 + 17.8T + 1.36e3T^{2} \) |
| 41 | \( 1 - 28.8T + 1.68e3T^{2} \) |
| 43 | \( 1 + 56.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 49.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 60.1T + 2.80e3T^{2} \) |
| 59 | \( 1 - 110. iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 34.2T + 3.72e3T^{2} \) |
| 67 | \( 1 - 131. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 52.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 62.2T + 5.32e3T^{2} \) |
| 79 | \( 1 - 103. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 57.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 145.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 66.7T + 9.40e3T^{2} \) |
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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
−10.14736817522543388409903998073, −9.113319766840770994605821248500, −8.426555352773267211682546629318, −7.65045464607122521964566982571, −6.85126331760935262788031875169, −5.66979924903387281835459453490, −5.35880708238194380066674902731, −3.79653612468405973352400898454, −2.41384824520753311564624444434, −0.958004429129846415978260664395,
0.52074370014220953197660033678, 2.04840118803009012007030067470, 2.89145564257107031192744820969, 4.30117998812513524580046069184, 4.96244406980201226585560885930, 6.65524722493179377446089987330, 7.37500461627919645570223942365, 7.965985282401344039222877101553, 9.245484381647021224392247406228, 9.665094289732319241519042920101
