| L(s) = 1 | + 0.618·2-s − 1.61·4-s + 5-s − 2.23·8-s + 0.618·10-s − 4.23·11-s − 0.763·13-s + 1.85·16-s + 5·17-s + 5.23·19-s − 1.61·20-s − 2.61·22-s + 6.47·23-s − 4·25-s − 0.472·26-s − 7.47·29-s − 8.70·31-s + 5.61·32-s + 3.09·34-s − 6·37-s + 3.23·38-s − 2.23·40-s + 1.70·41-s − 10·43-s + 6.85·44-s + 4.00·46-s − 7.70·47-s + ⋯ |
| L(s) = 1 | + 0.437·2-s − 0.809·4-s + 0.447·5-s − 0.790·8-s + 0.195·10-s − 1.27·11-s − 0.211·13-s + 0.463·16-s + 1.21·17-s + 1.20·19-s − 0.361·20-s − 0.558·22-s + 1.34·23-s − 0.800·25-s − 0.0925·26-s − 1.38·29-s − 1.56·31-s + 0.993·32-s + 0.529·34-s − 0.986·37-s + 0.524·38-s − 0.353·40-s + 0.266·41-s − 1.52·43-s + 1.03·44-s + 0.589·46-s − 1.12·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 239 | \( 1 - T \) |
| good | 2 | \( 1 - 0.618T + 2T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 4.23T + 11T^{2} \) |
| 13 | \( 1 + 0.763T + 13T^{2} \) |
| 17 | \( 1 - 5T + 17T^{2} \) |
| 19 | \( 1 - 5.23T + 19T^{2} \) |
| 23 | \( 1 - 6.47T + 23T^{2} \) |
| 29 | \( 1 + 7.47T + 29T^{2} \) |
| 31 | \( 1 + 8.70T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 - 1.70T + 41T^{2} \) |
| 43 | \( 1 + 10T + 43T^{2} \) |
| 47 | \( 1 + 7.70T + 47T^{2} \) |
| 53 | \( 1 + 2.47T + 53T^{2} \) |
| 59 | \( 1 + 3.23T + 59T^{2} \) |
| 61 | \( 1 + 3.94T + 61T^{2} \) |
| 67 | \( 1 + 6.23T + 67T^{2} \) |
| 71 | \( 1 - 4.94T + 71T^{2} \) |
| 73 | \( 1 + 2.94T + 73T^{2} \) |
| 79 | \( 1 - 2.94T + 79T^{2} \) |
| 83 | \( 1 - 4.23T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 + 8.18T + 97T^{2} \) |
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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
−8.773249679832803308630529509095, −7.82384885836759838679038909087, −7.31188957130062621483446400089, −6.00497004734199186396405818940, −5.18906760764357685328972627854, −5.08642092769509010734773371726, −3.56746990778968619026552713172, −3.07521573594279267283681992298, −1.61598418525318319117623378766, 0,
1.61598418525318319117623378766, 3.07521573594279267283681992298, 3.56746990778968619026552713172, 5.08642092769509010734773371726, 5.18906760764357685328972627854, 6.00497004734199186396405818940, 7.31188957130062621483446400089, 7.82384885836759838679038909087, 8.773249679832803308630529509095
