L(s) = 1 | − 2-s + 3-s + 4-s − 3·5-s − 6-s − 8-s + 9-s + 3·10-s + 2·11-s + 12-s + 7·13-s − 3·15-s + 16-s − 4·17-s − 18-s + 6·19-s − 3·20-s − 2·22-s + 23-s − 24-s + 4·25-s − 7·26-s + 27-s + 7·29-s + 3·30-s + 8·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.948·10-s + 0.603·11-s + 0.288·12-s + 1.94·13-s − 0.774·15-s + 1/4·16-s − 0.970·17-s − 0.235·18-s + 1.37·19-s − 0.670·20-s − 0.426·22-s + 0.208·23-s − 0.204·24-s + 4/5·25-s − 1.37·26-s + 0.192·27-s + 1.29·29-s + 0.547·30-s + 1.43·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.633819945\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.633819945\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−8.279571378287271087367947651558, −7.41028094761450276219524006505, −6.74584763263629197035459006245, −6.23111370989478572589184447338, −4.99734159864748113992778025863, −4.11827262294391537210205574478, −3.50992459723965618478835728160, −2.89587358693314809563243649723, −1.54463223673405186400982730406, −0.76814717640230417808143450932,
0.76814717640230417808143450932, 1.54463223673405186400982730406, 2.89587358693314809563243649723, 3.50992459723965618478835728160, 4.11827262294391537210205574478, 4.99734159864748113992778025863, 6.23111370989478572589184447338, 6.74584763263629197035459006245, 7.41028094761450276219524006505, 8.279571378287271087367947651558