L(s) = 1 | − 1.31·2-s − 3-s − 0.280·4-s + 1.11·5-s + 1.31·6-s + 2.99·8-s + 9-s − 1.46·10-s + 2.92·11-s + 0.280·12-s − 0.149·13-s − 1.11·15-s − 3.36·16-s + 0.809·17-s − 1.31·18-s + 7.67·19-s − 0.313·20-s − 3.83·22-s − 23-s − 2.99·24-s − 3.74·25-s + 0.196·26-s − 27-s + 9.97·29-s + 1.46·30-s + 1.00·31-s − 1.57·32-s + ⋯ |
L(s) = 1 | − 0.927·2-s − 0.577·3-s − 0.140·4-s + 0.500·5-s + 0.535·6-s + 1.05·8-s + 0.333·9-s − 0.463·10-s + 0.881·11-s + 0.0808·12-s − 0.0415·13-s − 0.288·15-s − 0.840·16-s + 0.196·17-s − 0.309·18-s + 1.76·19-s − 0.0700·20-s − 0.817·22-s − 0.208·23-s − 0.610·24-s − 0.749·25-s + 0.0385·26-s − 0.192·27-s + 1.85·29-s + 0.267·30-s + 0.180·31-s − 0.278·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.064675577\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.064675577\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 2 | \( 1 + 1.31T + 2T^{2} \) |
| 5 | \( 1 - 1.11T + 5T^{2} \) |
| 11 | \( 1 - 2.92T + 11T^{2} \) |
| 13 | \( 1 + 0.149T + 13T^{2} \) |
| 17 | \( 1 - 0.809T + 17T^{2} \) |
| 19 | \( 1 - 7.67T + 19T^{2} \) |
| 29 | \( 1 - 9.97T + 29T^{2} \) |
| 31 | \( 1 - 1.00T + 31T^{2} \) |
| 37 | \( 1 - 4.97T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 + 0.722T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 + 8.38T + 53T^{2} \) |
| 59 | \( 1 + 6.45T + 59T^{2} \) |
| 61 | \( 1 + 8.06T + 61T^{2} \) |
| 67 | \( 1 - 5.48T + 67T^{2} \) |
| 71 | \( 1 + 15.6T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 - 3.61T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 + 3.14T + 97T^{2} \) |
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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
−8.862755226932830516697025487459, −7.76429069028899666756899304089, −7.40550883050995543355711029340, −6.33451604076962909251778166544, −5.75815192654459491406621772255, −4.77428980939420058086985333665, −4.12736472782488055895854608515, −2.88018319260492007037376280829, −1.51919451818873788452295873720, −0.815830964029018728311806663114,
0.815830964029018728311806663114, 1.51919451818873788452295873720, 2.88018319260492007037376280829, 4.12736472782488055895854608515, 4.77428980939420058086985333665, 5.75815192654459491406621772255, 6.33451604076962909251778166544, 7.40550883050995543355711029340, 7.76429069028899666756899304089, 8.862755226932830516697025487459