| L(s) = 1 | − i·2-s − i·3-s − 4-s + (−2 + i)5-s − 6-s + i·8-s − 9-s + (1 + 2i)10-s − 6·11-s + i·12-s + i·13-s + (1 + 2i)15-s + 16-s + i·18-s − 6·19-s + (2 − i)20-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (−0.894 + 0.447i)5-s − 0.408·6-s + 0.353i·8-s − 0.333·9-s + (0.316 + 0.632i)10-s − 1.80·11-s + 0.288i·12-s + 0.277i·13-s + (0.258 + 0.516i)15-s + 0.250·16-s + 0.235i·18-s − 1.37·19-s + (0.447 − 0.223i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (2 - i)T \) |
| 13 | \( 1 - iT \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 6T + 11T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 10T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−10.85240569269252447258421599262, −10.09877033016297426394059805825, −8.721114826406882882491544725395, −7.914469002022591089368447615768, −7.16534286090752305271621679780, −5.79275595864651550240912883168, −4.55528528910644154233824829377, −3.28416136422331593769413605176, −2.18171372071285739740026906105, 0,
2.90566405700736196445289808821, 4.36069336101473660250875034031, 4.98628006169101036492933345719, 6.15728573959655977856396757472, 7.47020225606372979519662389201, 8.262799860054498916787014023187, 8.834150179522979669111286677836, 10.27860822179830146277111877190, 10.70809765424301575703797741022
