| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 7-s − 8-s + 9-s + 10-s − 5·11-s + 12-s + 2·13-s − 14-s − 15-s + 16-s + 17-s − 18-s + 6·19-s − 20-s + 21-s + 5·22-s − 24-s + 25-s − 2·26-s + 27-s + 28-s + 9·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.50·11-s + 0.288·12-s + 0.554·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 1.37·19-s − 0.223·20-s + 0.218·21-s + 1.06·22-s − 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.188·28-s + 1.67·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.357911511\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.357911511\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−9.834984120388578443132202063584, −8.920412853598702280342036743430, −7.996250495262613210513502734457, −7.82941954654821095457274233859, −6.78836585403012800490438370459, −5.59293502558607875381724906508, −4.63614282079891585361220034930, −3.30932495254934594571503231214, −2.50040087393469294415480604150, −0.990622762009589145457824222733,
0.990622762009589145457824222733, 2.50040087393469294415480604150, 3.30932495254934594571503231214, 4.63614282079891585361220034930, 5.59293502558607875381724906508, 6.78836585403012800490438370459, 7.82941954654821095457274233859, 7.996250495262613210513502734457, 8.920412853598702280342036743430, 9.834984120388578443132202063584
