L(s) = 1 | + (0.587 − 1.01i)5-s + (2.35 + 1.21i)7-s + (1.44 − 0.835i)11-s + 1.14i·13-s + (−2.07 − 3.59i)17-s + (1.83 + 1.05i)19-s + (4.22 + 2.43i)23-s + (1.81 + 3.13i)25-s − 8.32i·29-s + (7.18 − 4.14i)31-s + (2.61 − 1.68i)35-s + (−2.19 + 3.81i)37-s − 2.67·41-s + 4.08·43-s + (−1.75 + 3.03i)47-s + ⋯ |
L(s) = 1 | + (0.262 − 0.454i)5-s + (0.889 + 0.457i)7-s + (0.436 − 0.251i)11-s + 0.317i·13-s + (−0.503 − 0.872i)17-s + (0.420 + 0.242i)19-s + (0.880 + 0.508i)23-s + (0.362 + 0.627i)25-s − 1.54i·29-s + (1.28 − 0.744i)31-s + (0.441 − 0.284i)35-s + (−0.361 + 0.626i)37-s − 0.417·41-s + 0.623·43-s + (−0.255 + 0.442i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.66897 - 0.142253i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.66897 - 0.142253i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.35 - 1.21i)T \) |
good | 5 | \( 1 + (-0.587 + 1.01i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.44 + 0.835i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 1.14iT - 13T^{2} \) |
| 17 | \( 1 + (2.07 + 3.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.83 - 1.05i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.22 - 2.43i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 8.32iT - 29T^{2} \) |
| 31 | \( 1 + (-7.18 + 4.14i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.19 - 3.81i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 2.67T + 41T^{2} \) |
| 43 | \( 1 - 4.08T + 43T^{2} \) |
| 47 | \( 1 + (1.75 - 3.03i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.59 - 0.922i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.98 - 5.17i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (12.8 + 7.39i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.22 + 7.31i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 5.48iT - 71T^{2} \) |
| 73 | \( 1 + (-0.846 + 0.488i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.56 - 9.63i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 + (6.29 - 10.9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 9.52iT - 97T^{2} \) |
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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
−11.20602940638662944490050797316, −9.806763313143015822508118357328, −9.132870235954272709803622312312, −8.303172056281834470265282555739, −7.34275075545100268153520874621, −6.17319871823436870379595857048, −5.17506320851837562015180144015, −4.34460039165140230327714363977, −2.75470584209464423122196322163, −1.33274100207022313202208626620,
1.43244516300709594128406511631, 2.91677079015088944312420510398, 4.27625715904540480369197363548, 5.19994869259487065667755046193, 6.50251636650283550174988077714, 7.19907789397750957035116552439, 8.322242133744037155993532664677, 9.053450197460829647027998949565, 10.43566685309517863699464302090, 10.66987635062403477622742542435