L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 5·7-s + 8-s + 9-s + 10-s + 3·11-s + 12-s + 5·14-s + 15-s + 16-s − 8·17-s + 18-s + 5·19-s + 20-s + 5·21-s + 3·22-s − 4·23-s + 24-s + 25-s + 27-s + 5·28-s − 4·29-s + 30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.88·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.904·11-s + 0.288·12-s + 1.33·14-s + 0.258·15-s + 1/4·16-s − 1.94·17-s + 0.235·18-s + 1.14·19-s + 0.223·20-s + 1.09·21-s + 0.639·22-s − 0.834·23-s + 0.204·24-s + 1/5·25-s + 0.192·27-s + 0.944·28-s − 0.742·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.691775916\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.691775916\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−8.112307483301415307127814775721, −7.55867226063472289503177634668, −6.78802725756493159579396175974, −5.97247336569515810153926221953, −5.17540714730654727967228871076, −4.40253623867499602391633360295, −4.03568412251960486989113090679, −2.74570820479882314726008675223, −1.96172696082048736423905659781, −1.32952563140635806706734780660,
1.32952563140635806706734780660, 1.96172696082048736423905659781, 2.74570820479882314726008675223, 4.03568412251960486989113090679, 4.40253623867499602391633360295, 5.17540714730654727967228871076, 5.97247336569515810153926221953, 6.78802725756493159579396175974, 7.55867226063472289503177634668, 8.112307483301415307127814775721