L(s) = 1 | − 3·5-s + 7-s − 6·11-s − 13-s + 3·17-s + 2·19-s + 4·25-s + 6·29-s + 4·31-s − 3·35-s + 7·37-s − 43-s + 3·47-s − 6·49-s + 18·55-s + 6·59-s − 8·61-s + 3·65-s + 14·67-s − 3·71-s + 2·73-s − 6·77-s − 8·79-s − 12·83-s − 9·85-s + 6·89-s − 91-s + ⋯ |
L(s) = 1 | − 1.34·5-s + 0.377·7-s − 1.80·11-s − 0.277·13-s + 0.727·17-s + 0.458·19-s + 4/5·25-s + 1.11·29-s + 0.718·31-s − 0.507·35-s + 1.15·37-s − 0.152·43-s + 0.437·47-s − 6/7·49-s + 2.42·55-s + 0.781·59-s − 1.02·61-s + 0.372·65-s + 1.71·67-s − 0.356·71-s + 0.234·73-s − 0.683·77-s − 0.900·79-s − 1.31·83-s − 0.976·85-s + 0.635·89-s − 0.104·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−7.65375399907768794141633334268, −7.14405343281603376779046692886, −6.11833801898662524972237565313, −5.23279360252017967296025712816, −4.76217154176687478405167442531, −3.98115227907858376568912816370, −3.05367690895031831326025019966, −2.51487092538000809654409490646, −1.05562487871414231772127155218, 0,
1.05562487871414231772127155218, 2.51487092538000809654409490646, 3.05367690895031831326025019966, 3.98115227907858376568912816370, 4.76217154176687478405167442531, 5.23279360252017967296025712816, 6.11833801898662524972237565313, 7.14405343281603376779046692886, 7.65375399907768794141633334268