L(s) = 1 | + 2i·2-s − 3-s − 2·4-s − 3i·5-s − 2i·6-s + 9-s + 6·10-s + 2·12-s + (−2 + 3i)13-s + 3i·15-s − 4·16-s + 2·17-s + 2i·18-s − i·19-s + 6i·20-s + ⋯ |
L(s) = 1 | + 1.41i·2-s − 0.577·3-s − 4-s − 1.34i·5-s − 0.816i·6-s + 0.333·9-s + 1.89·10-s + 0.577·12-s + (−0.554 + 0.832i)13-s + 0.774i·15-s − 16-s + 0.485·17-s + 0.471i·18-s − 0.229i·19-s + 1.34i·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.285770570\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.285770570\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + (2 - 3i)T \) |
good | 2 | \( 1 - 2iT - 2T^{2} \) |
| 5 | \( 1 + 3iT - 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + iT - 19T^{2} \) |
| 23 | \( 1 + T + 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 + 5iT - 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 - 10iT - 41T^{2} \) |
| 43 | \( 1 - 9T + 43T^{2} \) |
| 47 | \( 1 + 7iT - 47T^{2} \) |
| 53 | \( 1 - 9T + 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 9iT - 73T^{2} \) |
| 79 | \( 1 - 15T + 79T^{2} \) |
| 83 | \( 1 + 9iT - 83T^{2} \) |
| 89 | \( 1 - 9iT - 89T^{2} \) |
| 97 | \( 1 - 13iT - 97T^{2} \) |
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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
−9.139274165213903881443030928130, −8.396816085323018510806684138000, −7.67415592127711886938838407225, −6.94652469701368816482946697459, −6.09580089464808147574494596939, −5.42375402997333623750021283572, −4.74762446685886443044772850489, −4.11621459657269073200947826360, −2.20167833315090543384049477002, −0.70642298823061256464299133382,
0.875768824128965648868113036169, 2.26274293826635469980855964714, 3.00599022414469394383473264966, 3.74782198970013195280826461896, 4.83668459120693716324189018656, 5.84779482884666686093636514868, 6.76573539585390080024313172034, 7.36661984272332526512423578548, 8.430053636755945120544174864856, 9.616711707044460839933550669370