| L(s) = 1 | + (−0.190 − 2.22i)5-s + 4.70i·7-s + 1.04·11-s + 0.691i·13-s + 3.55i·17-s − 5.45·19-s − 7.25i·23-s + (−4.92 + 0.849i)25-s − 8.75·29-s + 1.80·31-s + (10.4 − 0.897i)35-s + 5.40i·37-s − 0.0329·41-s + 1.55i·43-s − 2.35i·47-s + ⋯ |
| L(s) = 1 | + (−0.0852 − 0.996i)5-s + 1.77i·7-s + 0.313·11-s + 0.191i·13-s + 0.862i·17-s − 1.25·19-s − 1.51i·23-s + (−0.985 + 0.169i)25-s − 1.62·29-s + 0.323·31-s + (1.77 − 0.151i)35-s + 0.887i·37-s − 0.00514·41-s + 0.237i·43-s − 0.344i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08228583954\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08228583954\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.190 + 2.22i)T \) |
| good | 7 | \( 1 - 4.70iT - 7T^{2} \) |
| 11 | \( 1 - 1.04T + 11T^{2} \) |
| 13 | \( 1 - 0.691iT - 13T^{2} \) |
| 17 | \( 1 - 3.55iT - 17T^{2} \) |
| 19 | \( 1 + 5.45T + 19T^{2} \) |
| 23 | \( 1 + 7.25iT - 23T^{2} \) |
| 29 | \( 1 + 8.75T + 29T^{2} \) |
| 31 | \( 1 - 1.80T + 31T^{2} \) |
| 37 | \( 1 - 5.40iT - 37T^{2} \) |
| 41 | \( 1 + 0.0329T + 41T^{2} \) |
| 43 | \( 1 - 1.55iT - 43T^{2} \) |
| 47 | \( 1 + 2.35iT - 47T^{2} \) |
| 53 | \( 1 + 5.03iT - 53T^{2} \) |
| 59 | \( 1 + 7.30T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 + 11.5iT - 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 + 11.7iT - 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 + 4.61iT - 83T^{2} \) |
| 89 | \( 1 - 4.59T + 89T^{2} \) |
| 97 | \( 1 + 12.4iT - 97T^{2} \) |
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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
−9.043914246296443062246613471510, −8.366983127795619705121717745275, −7.914070738455514831389111867260, −6.40204278907772393752915618563, −6.15676648122168937718519052037, −5.16887778983999561156136230043, −4.56658445478752734667699997506, −3.58717446866230584364215143641, −2.33781352692084875826075768339, −1.69922263629213441436883240154,
0.02409688574624036025438688320, 1.41321586747884936888581309236, 2.63318709548687682134700724720, 3.78387049704978854984808157703, 3.98307616229003007185597913714, 5.19945406135380040772757789089, 6.20630255696711257719379525392, 6.92268705979965290112096129779, 7.47551643844410227017743920693, 7.917121514943375540440272451217
