| L(s) = 1 | − 0.453·3-s − 3.09·5-s − 2.79·9-s − 3.54·11-s + 6.65·13-s + 1.40·15-s − 17-s − 3.06·19-s − 2.80·23-s + 4.59·25-s + 2.62·27-s + 1.12·29-s − 8.64·31-s + 1.60·33-s − 3.44·37-s − 3.01·39-s − 4.53·41-s − 10.2·43-s + 8.65·45-s + 6.83·47-s + 0.453·51-s − 6.24·53-s + 10.9·55-s + 1.39·57-s + 4.06·59-s + 12.4·61-s − 20.6·65-s + ⋯ |
| L(s) = 1 | − 0.261·3-s − 1.38·5-s − 0.931·9-s − 1.06·11-s + 1.84·13-s + 0.362·15-s − 0.242·17-s − 0.704·19-s − 0.585·23-s + 0.918·25-s + 0.505·27-s + 0.209·29-s − 1.55·31-s + 0.279·33-s − 0.567·37-s − 0.483·39-s − 0.708·41-s − 1.56·43-s + 1.29·45-s + 0.997·47-s + 0.0634·51-s − 0.857·53-s + 1.47·55-s + 0.184·57-s + 0.528·59-s + 1.59·61-s − 2.55·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6495880980\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6495880980\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 0.453T + 3T^{2} \) |
| 5 | \( 1 + 3.09T + 5T^{2} \) |
| 11 | \( 1 + 3.54T + 11T^{2} \) |
| 13 | \( 1 - 6.65T + 13T^{2} \) |
| 19 | \( 1 + 3.06T + 19T^{2} \) |
| 23 | \( 1 + 2.80T + 23T^{2} \) |
| 29 | \( 1 - 1.12T + 29T^{2} \) |
| 31 | \( 1 + 8.64T + 31T^{2} \) |
| 37 | \( 1 + 3.44T + 37T^{2} \) |
| 41 | \( 1 + 4.53T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 - 6.83T + 47T^{2} \) |
| 53 | \( 1 + 6.24T + 53T^{2} \) |
| 59 | \( 1 - 4.06T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 + 6.60T + 67T^{2} \) |
| 71 | \( 1 - 6.60T + 71T^{2} \) |
| 73 | \( 1 - 5.57T + 73T^{2} \) |
| 79 | \( 1 - 4.57T + 79T^{2} \) |
| 83 | \( 1 - 8.30T + 83T^{2} \) |
| 89 | \( 1 - 2.87T + 89T^{2} \) |
| 97 | \( 1 - 4.43T + 97T^{2} \) |
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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
−8.281371182598922559619715773548, −8.236388753181734590328500668889, −7.19009279548023835481797664768, −6.35728419678754612185715551094, −5.60634669039491350946232102272, −4.80311535919754489650313184130, −3.73154457695291260741851811484, −3.36722640436760921509794672440, −2.05236014544039079387587479111, −0.46589661966613965993927666612,
0.46589661966613965993927666612, 2.05236014544039079387587479111, 3.36722640436760921509794672440, 3.73154457695291260741851811484, 4.80311535919754489650313184130, 5.60634669039491350946232102272, 6.35728419678754612185715551094, 7.19009279548023835481797664768, 8.236388753181734590328500668889, 8.281371182598922559619715773548
