| L(s) = 1 | + 5-s + 3·7-s + 5·11-s + 5·13-s − 2·17-s − 4·19-s + 23-s − 4·25-s + 9·29-s + 31-s + 3·35-s + 6·37-s + 3·41-s + 43-s + 3·47-s + 2·49-s − 2·53-s + 5·55-s + 11·59-s − 7·61-s + 5·65-s − 67-s − 4·71-s − 2·73-s + 15·77-s − 79-s + 83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.13·7-s + 1.50·11-s + 1.38·13-s − 0.485·17-s − 0.917·19-s + 0.208·23-s − 4/5·25-s + 1.67·29-s + 0.179·31-s + 0.507·35-s + 0.986·37-s + 0.468·41-s + 0.152·43-s + 0.437·47-s + 2/7·49-s − 0.274·53-s + 0.674·55-s + 1.43·59-s − 0.896·61-s + 0.620·65-s − 0.122·67-s − 0.474·71-s − 0.234·73-s + 1.70·77-s − 0.112·79-s + 0.109·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.079522759\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.079522759\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−8.454757898626656373586987582368, −7.56181188193185867073442418392, −6.45199230636397187869091058695, −6.31013975624867551002513034266, −5.33786121487458775831252469996, −4.26981978498061497407447229251, −4.06259190786348821922530320854, −2.73701963861613842182098209195, −1.70277894891131980685730358879, −1.07897034386658295091693896561,
1.07897034386658295091693896561, 1.70277894891131980685730358879, 2.73701963861613842182098209195, 4.06259190786348821922530320854, 4.26981978498061497407447229251, 5.33786121487458775831252469996, 6.31013975624867551002513034266, 6.45199230636397187869091058695, 7.56181188193185867073442418392, 8.454757898626656373586987582368
