| L(s) = 1 | − 5-s + 2·11-s + 6·13-s + 2·17-s + 9·23-s + 25-s − 3·29-s − 2·31-s + 8·37-s + 5·41-s + 43-s + 8·47-s − 4·53-s − 2·55-s − 8·59-s − 7·61-s − 6·65-s − 3·67-s − 8·71-s − 14·73-s + 4·79-s − 83-s − 2·85-s + 13·89-s + 10·97-s − 3·101-s − 13·103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.603·11-s + 1.66·13-s + 0.485·17-s + 1.87·23-s + 1/5·25-s − 0.557·29-s − 0.359·31-s + 1.31·37-s + 0.780·41-s + 0.152·43-s + 1.16·47-s − 0.549·53-s − 0.269·55-s − 1.04·59-s − 0.896·61-s − 0.744·65-s − 0.366·67-s − 0.949·71-s − 1.63·73-s + 0.450·79-s − 0.109·83-s − 0.216·85-s + 1.37·89-s + 1.01·97-s − 0.298·101-s − 1.28·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.404972371\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.404972371\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 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
−7.60265309210168748557324710519, −7.24754606042977475866524116452, −6.19804777733531948381155865399, −5.93559560733180451896858820193, −4.88585213858731081823702412615, −4.17486071426728806710633979955, −3.48189472952148380660318901243, −2.83944500895331134800078455735, −1.51371629611851711164973131032, −0.833177335468306893688187874330,
0.833177335468306893688187874330, 1.51371629611851711164973131032, 2.83944500895331134800078455735, 3.48189472952148380660318901243, 4.17486071426728806710633979955, 4.88585213858731081823702412615, 5.93559560733180451896858820193, 6.19804777733531948381155865399, 7.24754606042977475866524116452, 7.60265309210168748557324710519
