L(s) = 1 | − 1.73·7-s − 1.73i·13-s − i·19-s + 5·25-s − 10.3·31-s − 12.1i·37-s − 8i·43-s − 4·49-s − 8.66i·61-s − 11i·67-s + 17·73-s − 12.1·79-s + 2.99i·91-s − 5·97-s − 19.0·103-s + ⋯ |
L(s) = 1 | − 0.654·7-s − 0.480i·13-s − 0.229i·19-s + 25-s − 1.86·31-s − 1.99i·37-s − 1.21i·43-s − 0.571·49-s − 1.10i·61-s − 1.34i·67-s + 1.98·73-s − 1.36·79-s + 0.314i·91-s − 0.507·97-s − 1.87·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9806630931\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9806630931\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 + 1.73T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 1.73iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 + 12.1iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 8.66iT - 61T^{2} \) |
| 67 | \( 1 + 11iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 17T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 5T + 97T^{2} \) |
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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
−9.183038896596718625423131606910, −8.366957009282920705326374854449, −7.39136429811562132691874265181, −6.79406965166221887969949098323, −5.79895799472135257364139125355, −5.11083587173936887938683770236, −3.91777257187625406307865820586, −3.14514390406080205955181449968, −1.99128604064616779946479360110, −0.37475913513593105720857592842,
1.37310772760674899507851260408, 2.71104094898758633722727685173, 3.59858725927547907429093926267, 4.59548474466280092342475655965, 5.51994544852649503507864242575, 6.46793066856069550189606681203, 7.03402446446241007336999289247, 8.010994969276627439504625143348, 8.837342607047575105370615687623, 9.540301032746740201980824517213