L(s) = 1 | − 2-s + 4-s + 4·7-s − 8-s + 6·11-s − 13-s − 4·14-s + 16-s − 6·17-s + 2·19-s − 6·22-s + 6·23-s + 26-s + 4·28-s + 6·29-s + 2·31-s − 32-s + 6·34-s − 2·37-s − 2·38-s + 6·41-s − 2·43-s + 6·44-s − 6·46-s − 12·47-s + 9·49-s − 52-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.51·7-s − 0.353·8-s + 1.80·11-s − 0.277·13-s − 1.06·14-s + 1/4·16-s − 1.45·17-s + 0.458·19-s − 1.27·22-s + 1.25·23-s + 0.196·26-s + 0.755·28-s + 1.11·29-s + 0.359·31-s − 0.176·32-s + 1.02·34-s − 0.328·37-s − 0.324·38-s + 0.937·41-s − 0.304·43-s + 0.904·44-s − 0.884·46-s − 1.75·47-s + 9/7·49-s − 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.065629325\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.065629325\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.293783197284835659172214503833, −7.42504190155135261733452215043, −6.76663879014300041843938736888, −6.26932704793846804327537579966, −5.04094678931139452464434765842, −4.59592927781843994850092134261, −3.67648775517614272135688663849, −2.52067707967945062798640980123, −1.63016260959130671203856032638, −0.930877391236806221106891514820,
0.930877391236806221106891514820, 1.63016260959130671203856032638, 2.52067707967945062798640980123, 3.67648775517614272135688663849, 4.59592927781843994850092134261, 5.04094678931139452464434765842, 6.26932704793846804327537579966, 6.76663879014300041843938736888, 7.42504190155135261733452215043, 8.293783197284835659172214503833