L(s) = 1 | − 1.41i·2-s − 1.41i·3-s − 1.00·4-s − 2.00·6-s + 7-s − 1.00·9-s − 1.41i·11-s + 1.41i·12-s + 1.41i·13-s − 1.41i·14-s − 0.999·16-s + 1.41i·17-s + 1.41i·18-s + 1.41i·19-s − 1.41i·21-s − 2.00·22-s + ⋯ |
L(s) = 1 | − 1.41i·2-s − 1.41i·3-s − 1.00·4-s − 2.00·6-s + 7-s − 1.00·9-s − 1.41i·11-s + 1.41i·12-s + 1.41i·13-s − 1.41i·14-s − 0.999·16-s + 1.41i·17-s + 1.41i·18-s + 1.41i·19-s − 1.41i·21-s − 2.00·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 643 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 643 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9998375976\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9998375976\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 643 | \( 1 + T \) |
good | 2 | \( 1 + 1.41iT - T^{2} \) |
| 3 | \( 1 + 1.41iT - T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( 1 + 1.41iT - T^{2} \) |
| 13 | \( 1 - 1.41iT - T^{2} \) |
| 17 | \( 1 - 1.41iT - T^{2} \) |
| 19 | \( 1 - 1.41iT - T^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 1.41iT - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 + 1.41iT - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 1.41iT - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 1.41iT - T^{2} \) |
| 79 | \( 1 - 1.41iT - T^{2} \) |
| 83 | \( 1 + T + T^{2} \) |
| 89 | \( 1 - T + T^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−10.83393620990277159714687536110, −9.640466990688581648898876419575, −8.398714907456713505586374525086, −8.089968740191711208081140194445, −6.69237568284240319215992247555, −6.00730813306307802748592261812, −4.42456215706889537249931396120, −3.35015415167390916554184317334, −1.90117538784150014303107356247, −1.42930776884257491251972020313,
2.66937142137420542033697244843, 4.36523896901306282917837482643, 4.96123848042881506906606405373, 5.46230021353657668148743583308, 6.97469559691433318812029989732, 7.57229765495950296068733716981, 8.588436502787797536151879370326, 9.295957548365579215927861060039, 10.21073057568331177393524905809, 10.94851248684766075150090021217