L(s) = 1 | − 1.67·2-s − 3-s + 0.800·4-s + 1.67·6-s + 2.75·7-s + 2.00·8-s + 9-s − 3.09·11-s − 0.800·12-s − 4.10·13-s − 4.60·14-s − 4.96·16-s − 4.07·17-s − 1.67·18-s + 5.75·19-s − 2.75·21-s + 5.17·22-s − 6.80·23-s − 2.00·24-s + 6.86·26-s − 27-s + 2.20·28-s + 1.81·29-s + 3.24·31-s + 4.28·32-s + 3.09·33-s + 6.81·34-s + ⋯ |
L(s) = 1 | − 1.18·2-s − 0.577·3-s + 0.400·4-s + 0.683·6-s + 1.04·7-s + 0.709·8-s + 0.333·9-s − 0.932·11-s − 0.230·12-s − 1.13·13-s − 1.23·14-s − 1.24·16-s − 0.987·17-s − 0.394·18-s + 1.32·19-s − 0.600·21-s + 1.10·22-s − 1.41·23-s − 0.409·24-s + 1.34·26-s − 0.192·27-s + 0.416·28-s + 0.336·29-s + 0.583·31-s + 0.757·32-s + 0.538·33-s + 1.16·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5662212132\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5662212132\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 47 | \( 1 + T \) |
good | 2 | \( 1 + 1.67T + 2T^{2} \) |
| 7 | \( 1 - 2.75T + 7T^{2} \) |
| 11 | \( 1 + 3.09T + 11T^{2} \) |
| 13 | \( 1 + 4.10T + 13T^{2} \) |
| 17 | \( 1 + 4.07T + 17T^{2} \) |
| 19 | \( 1 - 5.75T + 19T^{2} \) |
| 23 | \( 1 + 6.80T + 23T^{2} \) |
| 29 | \( 1 - 1.81T + 29T^{2} \) |
| 31 | \( 1 - 3.24T + 31T^{2} \) |
| 37 | \( 1 - 5.48T + 37T^{2} \) |
| 41 | \( 1 - 1.31T + 41T^{2} \) |
| 43 | \( 1 + 8.65T + 43T^{2} \) |
| 53 | \( 1 - 5.58T + 53T^{2} \) |
| 59 | \( 1 - 0.623T + 59T^{2} \) |
| 61 | \( 1 - 6.04T + 61T^{2} \) |
| 67 | \( 1 - 4.05T + 67T^{2} \) |
| 71 | \( 1 - 15.1T + 71T^{2} \) |
| 73 | \( 1 + 3.81T + 73T^{2} \) |
| 79 | \( 1 + 5.60T + 79T^{2} \) |
| 83 | \( 1 + 4.45T + 83T^{2} \) |
| 89 | \( 1 + 2.86T + 89T^{2} \) |
| 97 | \( 1 + 6.01T + 97T^{2} \) |
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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
−8.388824684906625987690414663302, −7.958198500902283601488224091628, −7.38003093125837917881133553803, −6.59649583289204431905391566601, −5.39941919213343748174341955232, −4.89884260508544672274251892863, −4.17518519348368343979703605777, −2.57191934234269333490244120546, −1.74815881852006001530409563499, −0.54841505058656182986900305142,
0.54841505058656182986900305142, 1.74815881852006001530409563499, 2.57191934234269333490244120546, 4.17518519348368343979703605777, 4.89884260508544672274251892863, 5.39941919213343748174341955232, 6.59649583289204431905391566601, 7.38003093125837917881133553803, 7.958198500902283601488224091628, 8.388824684906625987690414663302