L(s) = 1 | + 2-s + 3-s − 4-s − 5-s + 6-s + 2·7-s − 3·8-s + 9-s − 10-s − 12-s − 4·13-s + 2·14-s − 15-s − 16-s + 6·17-s + 18-s + 6·19-s + 20-s + 2·21-s + 4·23-s − 3·24-s + 25-s − 4·26-s + 27-s − 2·28-s − 6·29-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 + 0.755·7-s − 1.06·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 1.10·13-s + 0.534·14-s − 0.258·15-s − 1/4·16-s + 1.45·17-s + 0.235·18-s + 1.37·19-s + 0.223·20-s + 0.436·21-s + 0.834·23-s − 0.612·24-s + 1/5·25-s − 0.784·26-s + 0.192·27-s − 0.377·28-s − 1.11·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.565830142\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.565830142\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−9.337631156181529963848325161510, −8.386473357379436040152945592348, −7.69344890685142555942945577499, −7.12224324890981759228654811108, −5.65462395656023150861273258602, −5.12106858409169630942967149663, −4.31338610647448896952875925848, −3.42381310446072218205035911280, −2.64207358991310319726851351291, −1.01147500994564153503812445251,
1.01147500994564153503812445251, 2.64207358991310319726851351291, 3.42381310446072218205035911280, 4.31338610647448896952875925848, 5.12106858409169630942967149663, 5.65462395656023150861273258602, 7.12224324890981759228654811108, 7.69344890685142555942945577499, 8.386473357379436040152945592348, 9.337631156181529963848325161510