| L(s) = 1 | + 3-s + 2·5-s − 2·7-s + 9-s + 5·13-s + 2·15-s + 17-s − 7·19-s − 2·21-s + 4·23-s − 25-s + 27-s + 6·29-s + 8·31-s − 4·35-s − 2·37-s + 5·39-s − 43-s + 2·45-s − 7·47-s − 3·49-s + 51-s + 2·53-s − 7·57-s + 4·59-s − 8·61-s − 2·63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s − 0.755·7-s + 1/3·9-s + 1.38·13-s + 0.516·15-s + 0.242·17-s − 1.60·19-s − 0.436·21-s + 0.834·23-s − 1/5·25-s + 0.192·27-s + 1.11·29-s + 1.43·31-s − 0.676·35-s − 0.328·37-s + 0.800·39-s − 0.152·43-s + 0.298·45-s − 1.02·47-s − 3/7·49-s + 0.140·51-s + 0.274·53-s − 0.927·57-s + 0.520·59-s − 1.02·61-s − 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.785813047\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.785813047\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.80073372789811, −13.25393647376940, −12.94268475253516, −12.56692072963661, −11.80414588304332, −11.30206358260215, −10.61497829294178, −10.23206326048181, −9.921583288457240, −9.185017217448256, −8.895414636450493, −8.253873045619702, −8.055940627392454, −7.029004821373165, −6.542438723526842, −6.285735752222701, −5.771104567548548, −4.989185075470720, −4.379204287706206, −3.829686514318064, −3.080378373894586, −2.783358299433363, −1.935790090634849, −1.416071882983836, −0.5891439654883862,
0.5891439654883862, 1.416071882983836, 1.935790090634849, 2.783358299433363, 3.080378373894586, 3.829686514318064, 4.379204287706206, 4.989185075470720, 5.771104567548548, 6.285735752222701, 6.542438723526842, 7.029004821373165, 8.055940627392454, 8.253873045619702, 8.895414636450493, 9.185017217448256, 9.921583288457240, 10.23206326048181, 10.61497829294178, 11.30206358260215, 11.80414588304332, 12.56692072963661, 12.94268475253516, 13.25393647376940, 13.80073372789811
