| L(s) = 1 | − 2.39·2-s + 3-s + 3.71·4-s − 2.39·6-s + 1.32·7-s − 4.11·8-s + 9-s + 3·11-s + 3.71·12-s − 6.11·13-s − 3.17·14-s + 2.39·16-s − 0.672·17-s − 2.39·18-s − 5.43·19-s + 1.32·21-s − 7.17·22-s + 7.89·23-s − 4.11·24-s + 14.6·26-s + 27-s + 4.93·28-s − 29-s + 2.50·32-s + 3·33-s + 1.60·34-s + 3.71·36-s + ⋯ |
| L(s) = 1 | − 1.69·2-s + 0.577·3-s + 1.85·4-s − 0.976·6-s + 0.501·7-s − 1.45·8-s + 0.333·9-s + 0.904·11-s + 1.07·12-s − 1.69·13-s − 0.848·14-s + 0.597·16-s − 0.163·17-s − 0.563·18-s − 1.24·19-s + 0.289·21-s − 1.52·22-s + 1.64·23-s − 0.838·24-s + 2.86·26-s + 0.192·27-s + 0.932·28-s − 0.185·29-s + 0.442·32-s + 0.522·33-s + 0.275·34-s + 0.619·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9785209839\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9785209839\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 2.39T + 2T^{2} \) |
| 7 | \( 1 - 1.32T + 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + 6.11T + 13T^{2} \) |
| 17 | \( 1 + 0.672T + 17T^{2} \) |
| 19 | \( 1 + 5.43T + 19T^{2} \) |
| 23 | \( 1 - 7.89T + 23T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 3.89T + 37T^{2} \) |
| 41 | \( 1 - 4.32T + 41T^{2} \) |
| 43 | \( 1 - 8.45T + 43T^{2} \) |
| 47 | \( 1 + 4.76T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 - 3.21T + 59T^{2} \) |
| 61 | \( 1 - 7.43T + 61T^{2} \) |
| 67 | \( 1 - 1.10T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 + 9.20T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 - 0.889T + 83T^{2} \) |
| 89 | \( 1 + 2.33T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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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
−9.094643408536488392872081325239, −8.459932900895329354240406095715, −7.67669623247393556519258627825, −7.09860990898367678667105539683, −6.44737959372255806264854839653, −5.04168403094639554591215467887, −4.12035256243533783564754959823, −2.66993356258791298783687593554, −2.01969015414635785340615067494, −0.815274929991884740931393064079,
0.815274929991884740931393064079, 2.01969015414635785340615067494, 2.66993356258791298783687593554, 4.12035256243533783564754959823, 5.04168403094639554591215467887, 6.44737959372255806264854839653, 7.09860990898367678667105539683, 7.67669623247393556519258627825, 8.459932900895329354240406095715, 9.094643408536488392872081325239
