L(s) = 1 | + (0.366 − 0.366i)2-s + 0.732i·4-s + (0.707 + 0.707i)7-s + (0.633 + 0.633i)8-s − 1.93i·11-s + 0.517·14-s − 0.267·16-s + (−0.707 − 0.707i)22-s + (1.36 + 1.36i)23-s + (−0.517 + 0.517i)28-s − 0.517·29-s + (−0.732 + 0.732i)32-s + (0.707 + 0.707i)37-s + (−0.707 + 0.707i)43-s + 1.41·44-s + ⋯ |
L(s) = 1 | + (0.366 − 0.366i)2-s + 0.732i·4-s + (0.707 + 0.707i)7-s + (0.633 + 0.633i)8-s − 1.93i·11-s + 0.517·14-s − 0.267·16-s + (−0.707 − 0.707i)22-s + (1.36 + 1.36i)23-s + (−0.517 + 0.517i)28-s − 0.517·29-s + (−0.732 + 0.732i)32-s + (0.707 + 0.707i)37-s + (−0.707 + 0.707i)43-s + 1.41·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.279i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.279i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.485884790\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.485884790\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
good | 2 | \( 1 + (-0.366 + 0.366i)T - iT^{2} \) |
| 11 | \( 1 + 1.93iT - T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (-1.36 - 1.36i)T + iT^{2} \) |
| 29 | \( 1 + 0.517T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (1 + i)T + iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + (1.22 + 1.22i)T + iT^{2} \) |
| 71 | \( 1 + 0.517iT - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + iT - T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−9.467749500272361641522256933885, −8.732027534142188365150973230684, −8.162702432183213383110961345965, −7.46935756190973302915501864359, −6.26364429203702270049841510121, −5.41757852189730883755636194103, −4.67073225633363492875681856977, −3.40389665926964421697743674206, −2.96577581991373207098789945165, −1.58762772048819548559872769886,
1.29118949503361770277262403942, 2.35138953418986841020647102354, 4.07683681254580985032185196946, 4.66488846935814731353260807325, 5.27227908560859830265680886104, 6.47231161385453316241618176362, 7.13647540383736666066156323550, 7.65386737811237934607768840691, 8.877054618015891054823097796902, 9.676226835133049066723231484742