L(s) = 1 | − 2-s + 1.41i·3-s − 1.41i·6-s − 1.41i·7-s + 8-s − 1.00·9-s − 1.41i·11-s + 13-s + 1.41i·14-s − 16-s − 1.41i·17-s + 1.00·18-s + 19-s + 2.00·21-s + 1.41i·22-s − 23-s + ⋯ |
L(s) = 1 | − 2-s + 1.41i·3-s − 1.41i·6-s − 1.41i·7-s + 8-s − 1.00·9-s − 1.41i·11-s + 13-s + 1.41i·14-s − 16-s − 1.41i·17-s + 1.00·18-s + 19-s + 2.00·21-s + 1.41i·22-s − 23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 751 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 751 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5381923100\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5381923100\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 751 | \( 1 - T \) |
good | 2 | \( 1 + T + T^{2} \) |
| 3 | \( 1 - 1.41iT - T^{2} \) |
| 5 | \( 1 + T^{2} \) |
| 7 | \( 1 + 1.41iT - T^{2} \) |
| 11 | \( 1 + 1.41iT - T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( 1 + 1.41iT - T^{2} \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( 1 - 1.41iT - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( 1 - T + T^{2} \) |
| 67 | \( 1 + 1.41iT - T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 1.41iT - T^{2} \) |
| 83 | \( 1 + 1.41iT - T^{2} \) |
| 89 | \( 1 - T + T^{2} \) |
| 97 | \( 1 - T + T^{2} \) |
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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
−10.35651866957979720096358036271, −9.673928885295727625658486278934, −9.171626692946047428752880614050, −8.128082750215580302253367260231, −7.51355238980620300435290255142, −6.13439928409172316706261872110, −4.94707303497722757937535216993, −4.05723600085610586457565379139, −3.33661979652402563438587541061, −0.914073185358732862521733003437,
1.55412332810692622166085780454, 2.20845489641497744547796932775, 4.08468593395387771530151555318, 5.58255149040677268423848842995, 6.32261504508002799283594173611, 7.43538701570160883080469817692, 7.996058707554852262575217903770, 8.719758964126735204866411302666, 9.517163490604226734078892592058, 10.34261454167724309566002468702