L(s) = 1 | + (−0.707 + 0.707i)3-s + (0.707 + 0.707i)5-s + 7-s − 1.00i·9-s − 1.41·11-s − 1.00·15-s + (−0.707 + 0.707i)21-s + 1.00i·25-s + (0.707 + 0.707i)27-s + 1.41i·29-s + 2i·31-s + (1.00 − 1.00i)33-s + (0.707 + 0.707i)35-s + (0.707 − 0.707i)45-s + 49-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)3-s + (0.707 + 0.707i)5-s + 7-s − 1.00i·9-s − 1.41·11-s − 1.00·15-s + (−0.707 + 0.707i)21-s + 1.00i·25-s + (0.707 + 0.707i)27-s + 1.41i·29-s + 2i·31-s + (1.00 − 1.00i)33-s + (0.707 + 0.707i)35-s + (0.707 − 0.707i)45-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.072206507\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.072206507\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (-0.707 - 0.707i)T \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + 1.41T + T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 - 1.41iT - T^{2} \) |
| 31 | \( 1 - 2iT - T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 - 1.41T + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (1 - i)T - iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (1 + i)T + iT^{2} \) |
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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
−8.994915895508182652286987550188, −8.413990533093743169958261670883, −7.33883979179595479860129468274, −6.80239376771649808066518204739, −5.73871818715661879794652564928, −5.25172465915438293034445005208, −4.69338518132098741202540077866, −3.48058843131609695830604290793, −2.65255869716228727201989170998, −1.46198703715848259801516380414,
0.71425747431589861088705840492, 1.97447066623838541783359502470, 2.48006486479540814297196404105, 4.27448996492997754182826278394, 4.93316276681664265789389373808, 5.63868314630088957736969817426, 6.05781391937905996396494269251, 7.20333208870545555022205527603, 7.989851831097416505859762794504, 8.216049902046553089152368692111