L(s) = 1 | + (0.119 + 0.207i)5-s + (0.5 − 0.866i)7-s + (−2.56 + 4.43i)11-s + (2.44 + 4.23i)13-s − 3.70·17-s + 3.66·19-s + (3.71 + 6.42i)23-s + (2.47 − 4.28i)25-s + (1.73 − 3.00i)29-s + (0.358 + 0.621i)31-s + 0.239·35-s + 4.60·37-s + (2.80 + 4.85i)41-s + (−6.24 + 10.8i)43-s + (2.16 − 3.75i)47-s + ⋯ |
L(s) = 1 | + (0.0534 + 0.0926i)5-s + (0.188 − 0.327i)7-s + (−0.772 + 1.33i)11-s + (0.677 + 1.17i)13-s − 0.898·17-s + 0.839·19-s + (0.773 + 1.34i)23-s + (0.494 − 0.856i)25-s + (0.321 − 0.557i)29-s + (0.0644 + 0.111i)31-s + 0.0404·35-s + 0.756·37-s + (0.437 + 0.757i)41-s + (−0.952 + 1.64i)43-s + (0.316 − 0.548i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.514 - 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.514 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.25634 + 0.711393i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.25634 + 0.711393i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
good | 5 | \( 1 + (-0.119 - 0.207i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2.56 - 4.43i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.44 - 4.23i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 3.70T + 17T^{2} \) |
| 19 | \( 1 - 3.66T + 19T^{2} \) |
| 23 | \( 1 + (-3.71 - 6.42i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.73 + 3.00i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.358 - 0.621i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 4.60T + 37T^{2} \) |
| 41 | \( 1 + (-2.80 - 4.85i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (6.24 - 10.8i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.16 + 3.75i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 0.942T + 53T^{2} \) |
| 59 | \( 1 + (-3.78 - 6.56i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.75 - 4.77i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.330 - 0.571i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 + 3.66T + 73T^{2} \) |
| 79 | \( 1 + (-3.11 + 5.39i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.85 + 8.40i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 7.48T + 89T^{2} \) |
| 97 | \( 1 + (-8.57 + 14.8i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.42025710789329726133523570617, −9.657063140907581614188974628816, −8.881919400038181801351055309203, −7.76217992397914997323708762102, −7.08186787671249462552274176983, −6.18782733067054460431905045438, −4.88378869325293721319612278991, −4.25219191718013642822264924975, −2.80417647697559494220843152678, −1.54150933418756011890979973225,
0.791004022050534418949399291963, 2.62845087921865106053227183901, 3.49367245637221570521648778131, 4.99795303987920026365180855680, 5.63798806532771217405505561115, 6.62567718722735293587289019477, 7.78132867749850366365323948218, 8.561621865151810939024791016877, 9.109420378907196040878508470939, 10.55376424537297243837352093548