L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s + 4·11-s − 12-s − 2·13-s + 16-s + 2·17-s − 18-s + 4·19-s − 4·22-s + 8·23-s + 24-s + 2·26-s − 27-s + 6·29-s + 8·31-s − 32-s − 4·33-s − 2·34-s + 36-s + 2·37-s − 4·38-s + 2·39-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 1.20·11-s − 0.288·12-s − 0.554·13-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.917·19-s − 0.852·22-s + 1.66·23-s + 0.204·24-s + 0.392·26-s − 0.192·27-s + 1.11·29-s + 1.43·31-s − 0.176·32-s − 0.696·33-s − 0.342·34-s + 1/6·36-s + 0.328·37-s − 0.648·38-s + 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.477659902\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.477659902\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−7.76273613606117477579615317414, −7.28748890929725522196299023973, −6.50852273047693459179421113355, −6.08775566810929960458151070193, −5.04435483885601882808423083926, −4.52568543973992960616491067288, −3.38582978084269172950091309498, −2.68081884608995418067852068372, −1.37109143333440786106920131673, −0.806443807139041556038800189905,
0.806443807139041556038800189905, 1.37109143333440786106920131673, 2.68081884608995418067852068372, 3.38582978084269172950091309498, 4.52568543973992960616491067288, 5.04435483885601882808423083926, 6.08775566810929960458151070193, 6.50852273047693459179421113355, 7.28748890929725522196299023973, 7.76273613606117477579615317414