L(s) = 1 | + 1.41i·3-s + 5-s − 7-s − 1.00·9-s + 11-s + 1.41i·15-s + 17-s + 19-s − 1.41i·21-s − 1.41i·29-s + 1.41i·31-s + 1.41i·33-s − 35-s + 1.41i·37-s + 1.41i·41-s + ⋯ |
L(s) = 1 | + 1.41i·3-s + 5-s − 7-s − 1.00·9-s + 11-s + 1.41i·15-s + 17-s + 19-s − 1.41i·21-s − 1.41i·29-s + 1.41i·31-s + 1.41i·33-s − 35-s + 1.41i·37-s + 1.41i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.394108835\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.394108835\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 1.41iT - T^{2} \) |
| 5 | \( 1 - T + T^{2} \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + 1.41iT - T^{2} \) |
| 31 | \( 1 - 1.41iT - T^{2} \) |
| 37 | \( 1 - 1.41iT - T^{2} \) |
| 41 | \( 1 - 1.41iT - T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( 1 + 1.41iT - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T + T^{2} \) |
| 67 | \( 1 + 1.41iT - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 + 1.41iT - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + 1.41iT - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.684980502553304155216705643809, −8.922507338867491402626439943921, −7.968873330144470489835380398063, −6.70288708059089294993608278330, −6.19056472651110000227843988914, −5.31217614313115425253310201074, −4.62759204067765548397395609711, −3.46400379384153039385833720929, −3.13738320677443425497528763966, −1.53318185159870885012074061222,
1.06602819575941458445746130751, 1.93331361085573211956091496664, 2.96023488152114112165797315315, 3.89527742979712287928193362844, 5.52831608313672084366181938018, 5.86054244901229944584324310744, 6.78974174744660007977565909557, 7.13590312465581883611692345281, 8.037016542285944529528452076250, 9.046714716080296934919855591226