| L(s) = 1 | + 7-s − 3·9-s + 4·13-s + 4·17-s + 4·19-s + 8·23-s + 2·29-s − 8·31-s − 8·37-s + 6·41-s + 8·43-s + 8·47-s + 49-s − 4·59-s − 6·61-s − 3·63-s + 8·67-s + 12·71-s − 4·73-s − 4·79-s + 9·81-s − 10·89-s + 4·91-s − 12·97-s − 18·101-s − 8·103-s + 8·107-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 9-s + 1.10·13-s + 0.970·17-s + 0.917·19-s + 1.66·23-s + 0.371·29-s − 1.43·31-s − 1.31·37-s + 0.937·41-s + 1.21·43-s + 1.16·47-s + 1/7·49-s − 0.520·59-s − 0.768·61-s − 0.377·63-s + 0.977·67-s + 1.42·71-s − 0.468·73-s − 0.450·79-s + 81-s − 1.05·89-s + 0.419·91-s − 1.21·97-s − 1.79·101-s − 0.788·103-s + 0.773·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.596512071\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.596512071\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−10.76299377764594672711794414955, −9.388130908813774704499416349915, −8.799943201337645514593114856325, −7.891615195790016418554183287292, −7.00986091168002114154949592471, −5.76976129738844340692422505330, −5.23888608559624005399306226283, −3.77722189059138930157306360701, −2.84137895703240412834312405836, −1.16646170267796244428478406670,
1.16646170267796244428478406670, 2.84137895703240412834312405836, 3.77722189059138930157306360701, 5.23888608559624005399306226283, 5.76976129738844340692422505330, 7.00986091168002114154949592471, 7.891615195790016418554183287292, 8.799943201337645514593114856325, 9.388130908813774704499416349915, 10.76299377764594672711794414955
