| L(s) = 1 | − 1.41i·2-s − 2.82i·3-s − 2.00·4-s − 4.00·6-s − 5·7-s + 2.82i·8-s + 0.999·9-s + 5·11-s + 5.65i·12-s + 16.9i·13-s + 7.07i·14-s + 4.00·16-s + 25·17-s − 1.41i·18-s + 19·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.942i·3-s − 0.500·4-s − 0.666·6-s − 0.714·7-s + 0.353i·8-s + 0.111·9-s + 0.454·11-s + 0.471i·12-s + 1.30i·13-s + 0.505i·14-s + 0.250·16-s + 1.47·17-s − 0.0785i·18-s + 19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.856479629\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.856479629\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 5 | \( 1 \) |
| 19 | \( 1 - 19T \) |
| good | 3 | \( 1 + 2.82iT - 9T^{2} \) |
| 7 | \( 1 + 5T + 49T^{2} \) |
| 11 | \( 1 - 5T + 121T^{2} \) |
| 13 | \( 1 - 16.9iT - 169T^{2} \) |
| 17 | \( 1 - 25T + 289T^{2} \) |
| 23 | \( 1 - 10T + 529T^{2} \) |
| 29 | \( 1 - 42.4iT - 841T^{2} \) |
| 31 | \( 1 + 42.4iT - 961T^{2} \) |
| 37 | \( 1 - 25.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 42.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 5T + 1.84e3T^{2} \) |
| 47 | \( 1 + 5T + 2.20e3T^{2} \) |
| 53 | \( 1 - 25.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 84.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 95T + 3.72e3T^{2} \) |
| 67 | \( 1 + 110. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 25T + 5.32e3T^{2} \) |
| 79 | \( 1 - 42.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 130T + 6.88e3T^{2} \) |
| 89 | \( 1 + 127. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 16.9iT - 9.40e3T^{2} \) |
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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
−9.606118211966517555391292995941, −9.057951027605482450854393626332, −7.85856415149340777837361958078, −7.10323629130530849406010191659, −6.35660815330511520494790949949, −5.25493502609258273970947770829, −4.00575267983123906126216096879, −3.09610562105664041592078813662, −1.81087620187265806863556497774, −0.901759653858270507503553447342,
0.904124601707509486743504366612, 3.15039508326237657315216564920, 3.73160006627737018888837910483, 4.99162029946400373069319372937, 5.58486635381427848456731686036, 6.61555550510853959924711211578, 7.55934448452906461579296165048, 8.319956246973393030920616733755, 9.491819361336164171731264663708, 9.823569354500829352619363421614
