L(s) = 1 | + (−0.707 − 1.22i)3-s + (−1.41 + 2.44i)5-s + (0.500 − 0.866i)9-s + (−3 − 5.19i)11-s − 5.65·13-s + 4·15-s + (−0.707 − 1.22i)17-s + (2.12 − 3.67i)19-s + (−2 + 3.46i)23-s + (−1.49 − 2.59i)25-s − 5.65·27-s − 6·29-s + (−1.41 − 2.44i)31-s + (−4.24 + 7.34i)33-s + (−1 + 1.73i)37-s + ⋯ |
L(s) = 1 | + (−0.408 − 0.707i)3-s + (−0.632 + 1.09i)5-s + (0.166 − 0.288i)9-s + (−0.904 − 1.56i)11-s − 1.56·13-s + 1.03·15-s + (−0.171 − 0.297i)17-s + (0.486 − 0.842i)19-s + (−0.417 + 0.722i)23-s + (−0.299 − 0.519i)25-s − 1.08·27-s − 1.11·29-s + (−0.254 − 0.439i)31-s + (−0.738 + 1.27i)33-s + (−0.164 + 0.284i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0909074 - 0.396306i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0909074 - 0.396306i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.707 + 1.22i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (1.41 - 2.44i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (3 + 5.19i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 5.65T + 13T^{2} \) |
| 17 | \( 1 + (0.707 + 1.22i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.12 + 3.67i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + (1.41 + 2.44i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 1.41T + 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 + (-1.41 + 2.44i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1 - 1.73i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.707 + 1.22i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.24 + 7.34i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + (-4.94 - 8.57i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 1.41T + 83T^{2} \) |
| 89 | \( 1 + (-2.12 + 3.67i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−11.25573839343498767882635572304, −10.17358478639274393025805731454, −9.102684234014371110764090000994, −7.55106449737827924696010942922, −7.42701313224300727325491717403, −6.24794942489058609487013975854, −5.25003890598559086697583154493, −3.59508679550738896970450194428, −2.55717560065044704763151970441, −0.26504616000564844217160980858,
2.14735459663931401592074952638, 4.16015109145715895867057724911, 4.77618696874641628020926903761, 5.49735330688782335024454071003, 7.36088261506686262227045385060, 7.81035567736802631815157181275, 9.137598173728200722523397397228, 9.987808808399888875278882660263, 10.55791666536249735567333592151, 11.87806816748218999577328841638