L(s) = 1 | + 2.00·2-s + 2.03·4-s + 0.905·5-s + 0.0686·8-s + 1.81·10-s − 0.716·11-s − 13-s − 3.93·16-s + 2.35·17-s − 6.63·19-s + 1.84·20-s − 1.43·22-s − 3.75·23-s − 4.17·25-s − 2.00·26-s − 3.25·29-s − 1.57·31-s − 8.03·32-s + 4.72·34-s + 5.20·37-s − 13.3·38-s + 0.0621·40-s + 4.92·41-s − 9.43·43-s − 1.45·44-s − 7.55·46-s − 8.31·47-s + ⋯ |
L(s) = 1 | + 1.42·2-s + 1.01·4-s + 0.405·5-s + 0.0242·8-s + 0.575·10-s − 0.215·11-s − 0.277·13-s − 0.982·16-s + 0.570·17-s − 1.52·19-s + 0.411·20-s − 0.306·22-s − 0.783·23-s − 0.835·25-s − 0.393·26-s − 0.604·29-s − 0.282·31-s − 1.41·32-s + 0.810·34-s + 0.856·37-s − 2.16·38-s + 0.00982·40-s + 0.768·41-s − 1.43·43-s − 0.219·44-s − 1.11·46-s − 1.21·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 - 2.00T + 2T^{2} \) |
| 5 | \( 1 - 0.905T + 5T^{2} \) |
| 11 | \( 1 + 0.716T + 11T^{2} \) |
| 17 | \( 1 - 2.35T + 17T^{2} \) |
| 19 | \( 1 + 6.63T + 19T^{2} \) |
| 23 | \( 1 + 3.75T + 23T^{2} \) |
| 29 | \( 1 + 3.25T + 29T^{2} \) |
| 31 | \( 1 + 1.57T + 31T^{2} \) |
| 37 | \( 1 - 5.20T + 37T^{2} \) |
| 41 | \( 1 - 4.92T + 41T^{2} \) |
| 43 | \( 1 + 9.43T + 43T^{2} \) |
| 47 | \( 1 + 8.31T + 47T^{2} \) |
| 53 | \( 1 + 14.0T + 53T^{2} \) |
| 59 | \( 1 - 0.716T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 - 9.39T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 - 3.47T + 73T^{2} \) |
| 79 | \( 1 - 13.0T + 79T^{2} \) |
| 83 | \( 1 - 3.54T + 83T^{2} \) |
| 89 | \( 1 - 12.0T + 89T^{2} \) |
| 97 | \( 1 + 7.43T + 97T^{2} \) |
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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
−7.72728297455021272409444477488, −6.59759847227891055396386772791, −6.26143016879574162564006219494, −5.48730891690743740477341980765, −4.88780172891742430865971181148, −4.08608028057408515302342018459, −3.48329397824332030959435470979, −2.48313548508561150960925188088, −1.82469829531909674316880208360, 0,
1.82469829531909674316880208360, 2.48313548508561150960925188088, 3.48329397824332030959435470979, 4.08608028057408515302342018459, 4.88780172891742430865971181148, 5.48730891690743740477341980765, 6.26143016879574162564006219494, 6.59759847227891055396386772791, 7.72728297455021272409444477488