L(s) = 1 | − 2-s − 2·3-s − 4-s + 2·6-s − 7-s + 3·8-s + 9-s + 6·11-s + 2·12-s − 13-s + 14-s − 16-s − 2·17-s − 18-s + 2·21-s − 6·22-s − 6·24-s + 26-s + 4·27-s + 28-s − 2·29-s + 8·31-s − 5·32-s − 12·33-s + 2·34-s − 36-s + 2·37-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s + 0.816·6-s − 0.377·7-s + 1.06·8-s + 1/3·9-s + 1.80·11-s + 0.577·12-s − 0.277·13-s + 0.267·14-s − 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.436·21-s − 1.27·22-s − 1.22·24-s + 0.196·26-s + 0.769·27-s + 0.188·28-s − 0.371·29-s + 1.43·31-s − 0.883·32-s − 2.08·33-s + 0.342·34-s − 1/6·36-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5424447633\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5424447633\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−9.041388901032921210749386467525, −8.488564193190309696061103974889, −7.43512544039986337924780473243, −6.53217983956817693895130257981, −6.15358755651542175304974210581, −4.94894839356977478697964244896, −4.41398973214380312193601069355, −3.35590189539502452177466361398, −1.64442088846378370182745669087, −0.59276282801906835055915123782,
0.59276282801906835055915123782, 1.64442088846378370182745669087, 3.35590189539502452177466361398, 4.41398973214380312193601069355, 4.94894839356977478697964244896, 6.15358755651542175304974210581, 6.53217983956817693895130257981, 7.43512544039986337924780473243, 8.488564193190309696061103974889, 9.041388901032921210749386467525