L(s) = 1 | + 0.233·2-s − 2.00i·3-s − 1.94·4-s + 5-s − 0.467i·6-s + 1.43i·7-s − 0.921·8-s − 1.00·9-s + 0.233·10-s + (2.74 − 1.86i)11-s + 3.89i·12-s − 5.35·13-s + 0.334i·14-s − 2.00i·15-s + 3.67·16-s − 4.91i·17-s + ⋯ |
L(s) = 1 | + 0.165·2-s − 1.15i·3-s − 0.972·4-s + 0.447·5-s − 0.190i·6-s + 0.541i·7-s − 0.325·8-s − 0.335·9-s + 0.0738·10-s + (0.826 − 0.562i)11-s + 1.12i·12-s − 1.48·13-s + 0.0894i·14-s − 0.516i·15-s + 0.918·16-s − 1.19i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0570i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0570i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7463105915\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7463105915\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 11 | \( 1 + (-2.74 + 1.86i)T \) |
| 19 | \( 1 + (3.73 + 2.24i)T \) |
good | 2 | \( 1 - 0.233T + 2T^{2} \) |
| 3 | \( 1 + 2.00iT - 3T^{2} \) |
| 7 | \( 1 - 1.43iT - 7T^{2} \) |
| 13 | \( 1 + 5.35T + 13T^{2} \) |
| 17 | \( 1 + 4.91iT - 17T^{2} \) |
| 23 | \( 1 + 4.98T + 23T^{2} \) |
| 29 | \( 1 + 4.09T + 29T^{2} \) |
| 31 | \( 1 + 3.87iT - 31T^{2} \) |
| 37 | \( 1 - 5.67iT - 37T^{2} \) |
| 41 | \( 1 + 5.79T + 41T^{2} \) |
| 43 | \( 1 - 9.77iT - 43T^{2} \) |
| 47 | \( 1 + 9.64T + 47T^{2} \) |
| 53 | \( 1 + 6.67iT - 53T^{2} \) |
| 59 | \( 1 + 3.93iT - 59T^{2} \) |
| 61 | \( 1 + 9.62iT - 61T^{2} \) |
| 67 | \( 1 - 12.3iT - 67T^{2} \) |
| 71 | \( 1 - 1.57iT - 71T^{2} \) |
| 73 | \( 1 - 9.80iT - 73T^{2} \) |
| 79 | \( 1 - 1.60T + 79T^{2} \) |
| 83 | \( 1 + 8.47iT - 83T^{2} \) |
| 89 | \( 1 + 16.0iT - 89T^{2} \) |
| 97 | \( 1 + 13.9iT - 97T^{2} \) |
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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
−9.571569885805696879194456598229, −8.627135099651427601225879180879, −7.893315909937034167662973555844, −6.90881483066821229090619264165, −6.17449386150867049545444555057, −5.21889270567187233702200578304, −4.35419454403472805363303239880, −2.91633128573560078903178881644, −1.83687521508110468339959845319, −0.31722691555061175144425540385,
1.86528330464251608188620520573, 3.66244968216804581515779754222, 4.15242282430359940329907213220, 4.90747870713527712756936063830, 5.79234811403528345646334039909, 6.92933992661888724906079958089, 8.002172492613445404007331808884, 9.024284954846415228100835509746, 9.542761279138665643624803703464, 10.31363274024432795474576443090