L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 3·7-s − 8-s + 9-s + 5·11-s + 12-s + 6·13-s + 3·14-s + 16-s + 4·17-s − 18-s + 5·19-s − 3·21-s − 5·22-s − 5·23-s − 24-s − 6·26-s + 27-s − 3·28-s + 8·29-s − 31-s − 32-s + 5·33-s − 4·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s + 1.50·11-s + 0.288·12-s + 1.66·13-s + 0.801·14-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 1.14·19-s − 0.654·21-s − 1.06·22-s − 1.04·23-s − 0.204·24-s − 1.17·26-s + 0.192·27-s − 0.566·28-s + 1.48·29-s − 0.179·31-s − 0.176·32-s + 0.870·33-s − 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.988620921\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.988620921\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 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
−8.469223761987985640528200747851, −7.71409728597758760015187983724, −6.86600071812334324498031541110, −6.29310821075383939804618348889, −5.75245586042601416746922238771, −4.29403131959025027622941786138, −3.42927483240036584364020604956, −3.12961954616398103678179743652, −1.65455188430504574805931236028, −0.923779304855118682796140617952,
0.923779304855118682796140617952, 1.65455188430504574805931236028, 3.12961954616398103678179743652, 3.42927483240036584364020604956, 4.29403131959025027622941786138, 5.75245586042601416746922238771, 6.29310821075383939804618348889, 6.86600071812334324498031541110, 7.71409728597758760015187983724, 8.469223761987985640528200747851