| L(s) = 1 | + 5-s − 2·7-s + 3·11-s − 6·13-s − 5·17-s + 2·19-s + 23-s + 25-s + 9·29-s − 2·35-s − 9·37-s − 8·41-s + 7·43-s + 8·47-s − 3·49-s + 2·53-s + 3·55-s − 3·59-s + 6·61-s − 6·65-s − 11·67-s + 6·71-s − 2·73-s − 6·77-s + 13·79-s − 6·83-s − 5·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s + 0.904·11-s − 1.66·13-s − 1.21·17-s + 0.458·19-s + 0.208·23-s + 1/5·25-s + 1.67·29-s − 0.338·35-s − 1.47·37-s − 1.24·41-s + 1.06·43-s + 1.16·47-s − 3/7·49-s + 0.274·53-s + 0.404·55-s − 0.390·59-s + 0.768·61-s − 0.744·65-s − 1.34·67-s + 0.712·71-s − 0.234·73-s − 0.683·77-s + 1.46·79-s − 0.658·83-s − 0.542·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.491554431\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.491554431\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−14.04875435314990, −13.64265817971004, −12.99643754952464, −12.45975161832326, −12.11022737824524, −11.70228235721987, −10.99628345765435, −10.28417414213436, −10.12089115451390, −9.424415610907706, −9.078563613899154, −8.625203259353199, −7.894966489678333, −7.025796369654846, −6.955827059130731, −6.394580405366401, −5.754704502978865, −5.040290847600037, −4.654606126853881, −3.984228687972742, −3.252082950958114, −2.650289409884618, −2.143318778703163, −1.311852366588254, −0.4010763270407729,
0.4010763270407729, 1.311852366588254, 2.143318778703163, 2.650289409884618, 3.252082950958114, 3.984228687972742, 4.654606126853881, 5.040290847600037, 5.754704502978865, 6.394580405366401, 6.955827059130731, 7.025796369654846, 7.894966489678333, 8.625203259353199, 9.078563613899154, 9.424415610907706, 10.12089115451390, 10.28417414213436, 10.99628345765435, 11.70228235721987, 12.11022737824524, 12.45975161832326, 12.99643754952464, 13.64265817971004, 14.04875435314990
