L(s) = 1 | + i·2-s + 2.59i·3-s − 4-s + 3.53i·5-s − 2.59·6-s − i·8-s − 3.73·9-s − 3.53·10-s + (−2.29 − 2.39i)11-s − 2.59i·12-s + 5.01·13-s − 9.18·15-s + 16-s − 3.89·17-s − 3.73i·18-s − 4.65·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 1.49i·3-s − 0.5·4-s + 1.58i·5-s − 1.05·6-s − 0.353i·8-s − 1.24·9-s − 1.11·10-s + (−0.691 − 0.722i)11-s − 0.749i·12-s + 1.39·13-s − 2.37·15-s + 0.250·16-s − 0.944·17-s − 0.880i·18-s − 1.06·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0929 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0929 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9505748666\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9505748666\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (2.29 + 2.39i)T \) |
good | 3 | \( 1 - 2.59iT - 3T^{2} \) |
| 5 | \( 1 - 3.53iT - 5T^{2} \) |
| 13 | \( 1 - 5.01T + 13T^{2} \) |
| 17 | \( 1 + 3.89T + 17T^{2} \) |
| 19 | \( 1 + 4.65T + 19T^{2} \) |
| 23 | \( 1 + 1.55T + 23T^{2} \) |
| 29 | \( 1 - 0.100iT - 29T^{2} \) |
| 31 | \( 1 + 0.279iT - 31T^{2} \) |
| 37 | \( 1 - 0.705T + 37T^{2} \) |
| 41 | \( 1 + 2.94T + 41T^{2} \) |
| 43 | \( 1 - 9.03iT - 43T^{2} \) |
| 47 | \( 1 + 6.56iT - 47T^{2} \) |
| 53 | \( 1 - 5.54T + 53T^{2} \) |
| 59 | \( 1 - 14.5iT - 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 - 3.98T + 67T^{2} \) |
| 71 | \( 1 + 8.45T + 71T^{2} \) |
| 73 | \( 1 - 3.89T + 73T^{2} \) |
| 79 | \( 1 - 7.69iT - 79T^{2} \) |
| 83 | \( 1 + 2.87T + 83T^{2} \) |
| 89 | \( 1 - 5.94iT - 89T^{2} \) |
| 97 | \( 1 - 16.7iT - 97T^{2} \) |
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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
−10.59739504690659069369552966405, −9.774194291735241392970101756982, −8.748508854553918545949352335969, −8.212367647782665426369671571474, −6.94746381891821354289195869070, −6.23319758649646776497272919589, −5.49795497516921102363306002781, −4.24985005262363860645089821518, −3.62983317916439579912238055163, −2.64669639839000638378168447601,
0.42182088750560705204079943112, 1.56717421279204087130398551983, 2.25482807113572142830130905864, 3.94035179205739497758979327305, 4.85822194331716285513305229977, 5.85063719646376929737277687267, 6.75648674646621887295117196758, 7.88054858522714119664394900547, 8.510078998966810547865965120495, 8.968758418907028410877901785101