| L(s) = 1 | + (0.809 − 0.587i)2-s + (0.671 − 2.06i)3-s + (0.309 − 0.951i)4-s + (−0.284 − 2.21i)5-s + (−0.671 − 2.06i)6-s + 7-s + (−0.309 − 0.951i)8-s + (−1.38 − 1.00i)9-s + (−1.53 − 1.62i)10-s + (−4.04 + 2.93i)11-s + (−1.75 − 1.27i)12-s + (4.05 + 2.94i)13-s + (0.809 − 0.587i)14-s + (−4.77 − 0.900i)15-s + (−0.809 − 0.587i)16-s + (−0.205 − 0.633i)17-s + ⋯ |
| L(s) = 1 | + (0.572 − 0.415i)2-s + (0.387 − 1.19i)3-s + (0.154 − 0.475i)4-s + (−0.127 − 0.991i)5-s + (−0.274 − 0.843i)6-s + 0.377·7-s + (−0.109 − 0.336i)8-s + (−0.463 − 0.336i)9-s + (−0.485 − 0.514i)10-s + (−1.21 + 0.885i)11-s + (−0.507 − 0.368i)12-s + (1.12 + 0.816i)13-s + (0.216 − 0.157i)14-s + (−1.23 − 0.232i)15-s + (−0.202 − 0.146i)16-s + (−0.0499 − 0.153i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.532 + 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.532 + 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.972767 - 1.76095i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.972767 - 1.76095i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.809 + 0.587i)T \) |
| 5 | \( 1 + (0.284 + 2.21i)T \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + (-0.671 + 2.06i)T + (-2.42 - 1.76i)T^{2} \) |
| 11 | \( 1 + (4.04 - 2.93i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-4.05 - 2.94i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (0.205 + 0.633i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-1.26 - 3.90i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-1.55 + 1.13i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-2.49 + 7.67i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.881 - 2.71i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (7.72 + 5.61i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (0.985 + 0.715i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 8.81T + 43T^{2} \) |
| 47 | \( 1 + (1.69 - 5.20i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (4.01 - 12.3i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (8.07 + 5.86i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-8.36 + 6.07i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-0.633 - 1.94i)T + (-54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (-0.599 + 1.84i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.439 + 0.319i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (3.49 - 10.7i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-3.70 - 11.4i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (3.67 - 2.67i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-1.07 + 3.31i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−11.49190953199336630121700316996, −10.41383486047611336533824640197, −9.215533810181897256405477656730, −8.172778965571430614298343895572, −7.51542352241356190630296167352, −6.28118364287601044022181080752, −5.12437146216225955543387089419, −4.08950463072770090842998320603, −2.34386125212604484179889565057, −1.32664547521649338417714951239,
2.97057055049714657844574697532, 3.50422104363660947953037400598, 4.86229269017329762868252229044, 5.75247199694772696065363674374, 6.98976376958441632060738426257, 8.121561679187401810286555489955, 8.846893693387618576392814076343, 10.29058160175173222369472033712, 10.74917982223518429233726785200, 11.49593789048893831627101701190
