L(s) = 1 | − 2.11·2-s + 3-s + 2.45·4-s − 2.44·5-s − 2.11·6-s − 0.963·8-s + 9-s + 5.17·10-s + 1.45·11-s + 2.45·12-s − 1.37·13-s − 2.44·15-s − 2.87·16-s + 4.85·17-s − 2.11·18-s + 0.769·19-s − 6.01·20-s − 3.07·22-s − 1.47·23-s − 0.963·24-s + 0.998·25-s + 2.90·26-s + 27-s + 2.11·29-s + 5.17·30-s − 3.75·31-s + 8.00·32-s + ⋯ |
L(s) = 1 | − 1.49·2-s + 0.577·3-s + 1.22·4-s − 1.09·5-s − 0.861·6-s − 0.340·8-s + 0.333·9-s + 1.63·10-s + 0.439·11-s + 0.709·12-s − 0.381·13-s − 0.632·15-s − 0.719·16-s + 1.17·17-s − 0.497·18-s + 0.176·19-s − 1.34·20-s − 0.655·22-s − 0.306·23-s − 0.196·24-s + 0.199·25-s + 0.569·26-s + 0.192·27-s + 0.391·29-s + 0.943·30-s − 0.674·31-s + 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8103727516\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8103727516\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 2 | \( 1 + 2.11T + 2T^{2} \) |
| 5 | \( 1 + 2.44T + 5T^{2} \) |
| 11 | \( 1 - 1.45T + 11T^{2} \) |
| 13 | \( 1 + 1.37T + 13T^{2} \) |
| 17 | \( 1 - 4.85T + 17T^{2} \) |
| 19 | \( 1 - 0.769T + 19T^{2} \) |
| 23 | \( 1 + 1.47T + 23T^{2} \) |
| 29 | \( 1 - 2.11T + 29T^{2} \) |
| 31 | \( 1 + 3.75T + 31T^{2} \) |
| 37 | \( 1 - 4.91T + 37T^{2} \) |
| 43 | \( 1 + 1.59T + 43T^{2} \) |
| 47 | \( 1 - 4.11T + 47T^{2} \) |
| 53 | \( 1 + 5.74T + 53T^{2} \) |
| 59 | \( 1 - 3.21T + 59T^{2} \) |
| 61 | \( 1 - 7.29T + 61T^{2} \) |
| 67 | \( 1 + 13.8T + 67T^{2} \) |
| 71 | \( 1 - 6.70T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 - 0.262T + 79T^{2} \) |
| 83 | \( 1 - 6.38T + 83T^{2} \) |
| 89 | \( 1 + 4.85T + 89T^{2} \) |
| 97 | \( 1 + 1.57T + 97T^{2} \) |
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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
−8.041235167193280340629339019049, −7.68280114545359009208793502747, −7.13445882634374123428849598703, −6.31658029985132894822782228499, −5.17322069822136232210409595920, −4.21202414074788393712507578702, −3.54880313645151194948317476051, −2.59894727366147859649149253444, −1.55636247132475207625626571048, −0.60290985493430040258104422520,
0.60290985493430040258104422520, 1.55636247132475207625626571048, 2.59894727366147859649149253444, 3.54880313645151194948317476051, 4.21202414074788393712507578702, 5.17322069822136232210409595920, 6.31658029985132894822782228499, 7.13445882634374123428849598703, 7.68280114545359009208793502747, 8.041235167193280340629339019049