| L(s) = 1 | − 0.170·2-s − 1.97·4-s − 2.09·5-s + 7-s + 0.678·8-s + 0.357·10-s − 0.105·11-s + 0.808·13-s − 0.170·14-s + 3.82·16-s − 2.11·17-s + 0.0869·19-s + 4.12·20-s + 0.0179·22-s − 5.49·23-s − 0.621·25-s − 0.137·26-s − 1.97·28-s + 6.62·29-s + 5.84·31-s − 2.00·32-s + 0.361·34-s − 2.09·35-s + 7.11·37-s − 0.0148·38-s − 1.41·40-s − 12.0·41-s + ⋯ |
| L(s) = 1 | − 0.120·2-s − 0.985·4-s − 0.935·5-s + 0.377·7-s + 0.239·8-s + 0.112·10-s − 0.0316·11-s + 0.224·13-s − 0.0456·14-s + 0.956·16-s − 0.514·17-s + 0.0199·19-s + 0.922·20-s + 0.00382·22-s − 1.14·23-s − 0.124·25-s − 0.0270·26-s − 0.372·28-s + 1.23·29-s + 1.04·31-s − 0.355·32-s + 0.0620·34-s − 0.353·35-s + 1.17·37-s − 0.00240·38-s − 0.224·40-s − 1.87·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 + 0.170T + 2T^{2} \) |
| 5 | \( 1 + 2.09T + 5T^{2} \) |
| 11 | \( 1 + 0.105T + 11T^{2} \) |
| 13 | \( 1 - 0.808T + 13T^{2} \) |
| 17 | \( 1 + 2.11T + 17T^{2} \) |
| 19 | \( 1 - 0.0869T + 19T^{2} \) |
| 23 | \( 1 + 5.49T + 23T^{2} \) |
| 29 | \( 1 - 6.62T + 29T^{2} \) |
| 31 | \( 1 - 5.84T + 31T^{2} \) |
| 37 | \( 1 - 7.11T + 37T^{2} \) |
| 41 | \( 1 + 12.0T + 41T^{2} \) |
| 43 | \( 1 + 7.06T + 43T^{2} \) |
| 47 | \( 1 + 2.31T + 47T^{2} \) |
| 53 | \( 1 + 6.77T + 53T^{2} \) |
| 59 | \( 1 + 2.38T + 59T^{2} \) |
| 61 | \( 1 - 6.80T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 - 9.05T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 - 0.600T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 + 9.17T + 97T^{2} \) |
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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
−7.88338424767307163306864331150, −6.77794410734679331917777379976, −6.19756799028378898523653602406, −5.09270455925616266994100344236, −4.69353007860455351531204111286, −3.92704808150444204775408203577, −3.38318760919709705811698801722, −2.18610476813485004856096671079, −1.00390754946349407716120429494, 0,
1.00390754946349407716120429494, 2.18610476813485004856096671079, 3.38318760919709705811698801722, 3.92704808150444204775408203577, 4.69353007860455351531204111286, 5.09270455925616266994100344236, 6.19756799028378898523653602406, 6.77794410734679331917777379976, 7.88338424767307163306864331150
