L(s) = 1 | + 1.41·5-s − 1.41·13-s − 1.41·17-s − 4·23-s − 2.99·25-s − 6·29-s − 5.65·31-s − 4·37-s − 7.07·41-s + 4·43-s + 11.3·47-s − 4·53-s − 5.65·59-s − 4.24·61-s − 2.00·65-s + 4·67-s − 12·71-s + 7.07·73-s + 8·79-s − 5.65·83-s − 2.00·85-s + 12.7·89-s − 4.24·97-s − 9.89·101-s + 11.3·103-s − 16·107-s − 4·109-s + ⋯ |
L(s) = 1 | + 0.632·5-s − 0.392·13-s − 0.342·17-s − 0.834·23-s − 0.599·25-s − 1.11·29-s − 1.01·31-s − 0.657·37-s − 1.10·41-s + 0.609·43-s + 1.65·47-s − 0.549·53-s − 0.736·59-s − 0.543·61-s − 0.248·65-s + 0.488·67-s − 1.42·71-s + 0.827·73-s + 0.900·79-s − 0.620·83-s − 0.216·85-s + 1.34·89-s − 0.430·97-s − 0.985·101-s + 1.11·103-s − 1.54·107-s − 0.383·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 1.41T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 1.41T + 13T^{2} \) |
| 17 | \( 1 + 1.41T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 7.07T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 + 5.65T + 59T^{2} \) |
| 61 | \( 1 + 4.24T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 7.07T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 5.65T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 + 4.24T + 97T^{2} \) |
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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
−8.146710740537985992749354521380, −7.46976054255899795635940815642, −6.71159919778973700702533389428, −5.84246639401792291982404946958, −5.34297677225486722465064511045, −4.31152794951738830277880380339, −3.51168496479627164366990757352, −2.35840819735057846957579832260, −1.64811806557559284208977266929, 0,
1.64811806557559284208977266929, 2.35840819735057846957579832260, 3.51168496479627164366990757352, 4.31152794951738830277880380339, 5.34297677225486722465064511045, 5.84246639401792291982404946958, 6.71159919778973700702533389428, 7.46976054255899795635940815642, 8.146710740537985992749354521380