| L(s) = 1 | + i·2-s − 2i·3-s + 4-s + 2·6-s − i·7-s + 3i·8-s − 9-s + 11-s − 2i·12-s − 4i·13-s + 14-s − 16-s + 4i·17-s − i·18-s − 2·21-s + i·22-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 1.15i·3-s + 0.5·4-s + 0.816·6-s − 0.377i·7-s + 1.06i·8-s − 0.333·9-s + 0.301·11-s − 0.577i·12-s − 1.10i·13-s + 0.267·14-s − 0.250·16-s + 0.970i·17-s − 0.235i·18-s − 0.436·21-s + 0.213i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.266340606\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.266340606\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - iT - 2T^{2} \) |
| 3 | \( 1 + 2iT - 3T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 10T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 + 12iT - 43T^{2} \) |
| 47 | \( 1 + 10iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 2T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 8iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−8.620955911609377922595486244277, −8.174492283481955528778622374035, −7.36584827890653499760017117079, −6.96291241772005650928043398338, −6.08293364216541482051082461888, −5.58462429807133170957624296101, −4.29041873153501702364939041195, −3.03178462562410694041578916247, −2.01156619727075697395212295152, −0.947003665188110625140996323384,
1.23714716037308604460338377373, 2.57530640345614782397191181207, 3.21988218125237988527319976309, 4.50297198453242379290658397893, 4.64462093272092201867296886648, 6.22609583639507254366492295648, 6.66106332374348458884267154469, 7.77640365622775922741520520700, 8.836199784352373422992317708260, 9.536122300748006273467764064858
