L(s) = 1 | + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.707 − 0.707i)8-s + (−0.707 + 1.22i)11-s + (0.500 + 0.866i)16-s + (1 − 0.999i)22-s + (−1.22 + 0.707i)23-s + (0.5 − 0.866i)25-s + 1.41·29-s + (−0.258 − 0.965i)32-s + (−1.73 + i)37-s + 2i·43-s + (−1.22 + 0.707i)44-s + (1.36 − 0.366i)46-s + (−0.707 + 0.707i)50-s + ⋯ |
L(s) = 1 | + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.707 − 0.707i)8-s + (−0.707 + 1.22i)11-s + (0.500 + 0.866i)16-s + (1 − 0.999i)22-s + (−1.22 + 0.707i)23-s + (0.5 − 0.866i)25-s + 1.41·29-s + (−0.258 − 0.965i)32-s + (−1.73 + i)37-s + 2i·43-s + (−1.22 + 0.707i)44-s + (1.36 − 0.366i)46-s + (−0.707 + 0.707i)50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0431 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0431 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5578805483\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5578805483\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.965 + 0.258i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - 1.41T + T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 2iT - T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + 1.41iT - T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.971654631123085143836181133218, −8.046758810183412767842237998695, −7.78485438871308811721309773308, −6.78181964232729475453494471097, −6.28193706404268431563227937534, −5.09941900142558183448692871949, −4.29430360142847573782809473420, −3.14576159978038115445406645992, −2.34614984608526921491076286713, −1.38477225390658178363107636639,
0.44574129900738767493201276418, 1.84410225761136797398206048220, 2.82170661158180069355173817948, 3.70485468602407651253823154050, 5.09307503383877204095591975372, 5.66981935358381427318497402563, 6.49518610043787598660935069463, 7.15639787726413486360491526038, 8.045099428452050417847880308963, 8.536582891609522788642894999224