| L(s) = 1 | + 5-s − 3·7-s − 3·9-s − 6·11-s − 6·13-s − 17-s + 3·23-s − 4·25-s + 2·29-s − 3·31-s − 3·35-s − 2·37-s − 11·41-s + 6·43-s − 3·45-s + 6·47-s + 2·49-s + 2·53-s − 6·55-s + 6·59-s − 10·61-s + 9·63-s − 6·65-s + 3·67-s − 12·71-s − 9·73-s + 18·77-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.13·7-s − 9-s − 1.80·11-s − 1.66·13-s − 0.242·17-s + 0.625·23-s − 4/5·25-s + 0.371·29-s − 0.538·31-s − 0.507·35-s − 0.328·37-s − 1.71·41-s + 0.914·43-s − 0.447·45-s + 0.875·47-s + 2/7·49-s + 0.274·53-s − 0.809·55-s + 0.781·59-s − 1.28·61-s + 1.13·63-s − 0.744·65-s + 0.366·67-s − 1.42·71-s − 1.05·73-s + 2.05·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3135456640\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3135456640\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−7.73682961333853601575770446443, −7.18593538758924405218360984843, −6.44956786461568034262025539328, −5.47985199425574243066917793684, −5.38463007929820859164089841732, −4.38983157016923178447095446406, −3.14444493409304980244631566003, −2.76303777000775109266559464053, −2.07572988161597836029213598244, −0.24996371582930184464391225358,
0.24996371582930184464391225358, 2.07572988161597836029213598244, 2.76303777000775109266559464053, 3.14444493409304980244631566003, 4.38983157016923178447095446406, 5.38463007929820859164089841732, 5.47985199425574243066917793684, 6.44956786461568034262025539328, 7.18593538758924405218360984843, 7.73682961333853601575770446443
