L(s) = 1 | + (−2.13 + 3.70i)2-s + (−1.5 − 2.59i)3-s + (−5.15 − 8.92i)4-s + (2.5 − 4.33i)5-s + 12.8·6-s + (17.2 + 6.74i)7-s + 9.84·8-s + (−4.5 + 7.79i)9-s + (10.6 + 18.5i)10-s + (29.5 + 51.1i)11-s + (−15.4 + 26.7i)12-s − 60.4·13-s + (−61.8 + 49.4i)14-s − 15.0·15-s + (20.1 − 34.8i)16-s + (49.1 + 85.1i)17-s + ⋯ |
L(s) = 1 | + (−0.756 + 1.30i)2-s + (−0.288 − 0.499i)3-s + (−0.643 − 1.11i)4-s + (0.223 − 0.387i)5-s + 0.873·6-s + (0.931 + 0.364i)7-s + 0.435·8-s + (−0.166 + 0.288i)9-s + (0.338 + 0.585i)10-s + (0.808 + 1.40i)11-s + (−0.371 + 0.643i)12-s − 1.28·13-s + (−1.18 + 0.944i)14-s − 0.258·15-s + (0.314 − 0.545i)16-s + (0.701 + 1.21i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.672 - 0.740i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.672 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.366128 + 0.827316i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.366128 + 0.827316i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.5 + 2.59i)T \) |
| 5 | \( 1 + (-2.5 + 4.33i)T \) |
| 7 | \( 1 + (-17.2 - 6.74i)T \) |
good | 2 | \( 1 + (2.13 - 3.70i)T + (-4 - 6.92i)T^{2} \) |
| 11 | \( 1 + (-29.5 - 51.1i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 60.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-49.1 - 85.1i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-31.7 + 54.9i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (107. - 186. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 73.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-82.9 - 143. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (131. - 227. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 343.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 296.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-63.8 + 110. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (36.4 + 63.1i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (135. + 234. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-79.4 + 137. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (311. + 540. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 580.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-114. - 197. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-411. + 712. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 912.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-491. + 850. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 228.T + 9.12e5T^{2} \) |
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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
−14.11030250768992921173991881656, −12.45261455953030429802808294958, −11.82380462030297693629980262237, −10.00104379173506477939434625159, −9.123351812989353492282819076092, −7.85871603428042731150990487702, −7.23746622366735371161613826185, −5.89081504052555557016113551579, −4.83438310701449878636744108371, −1.63546013867335008644626599272,
0.71357183840031704398453938089, 2.59504994274735853723777519592, 4.10062791612587876313215087943, 5.81392107795364765191028444097, 7.68196139094098099928575005813, 8.965737574277155720743702473112, 9.925595426511282762762008795830, 10.76814672066100118332169303705, 11.58745313148160240828722170668, 12.27058995265735994915121490065