| L(s) = 1 | − 2.79·2-s + (0.351 + 1.69i)3-s + 5.80·4-s − i·5-s + (−0.983 − 4.73i)6-s + i·7-s − 10.6·8-s + (−2.75 + 1.19i)9-s + 2.79i·10-s + (1.01 − 3.15i)11-s + (2.04 + 9.83i)12-s − 1.61i·13-s − 2.79i·14-s + (1.69 − 0.351i)15-s + 18.0·16-s + 0.978·17-s + ⋯ |
| L(s) = 1 | − 1.97·2-s + (0.203 + 0.979i)3-s + 2.90·4-s − 0.447i·5-s + (−0.401 − 1.93i)6-s + 0.377i·7-s − 3.75·8-s + (−0.917 + 0.397i)9-s + 0.883i·10-s + (0.306 − 0.951i)11-s + (0.589 + 2.84i)12-s − 0.447i·13-s − 0.746i·14-s + (0.437 − 0.0908i)15-s + 4.51·16-s + 0.237·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.107i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2483284355\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2483284355\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.351 - 1.69i)T \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + (-1.01 + 3.15i)T \) |
| good | 2 | \( 1 + 2.79T + 2T^{2} \) |
| 13 | \( 1 + 1.61iT - 13T^{2} \) |
| 17 | \( 1 - 0.978T + 17T^{2} \) |
| 19 | \( 1 + 0.395iT - 19T^{2} \) |
| 23 | \( 1 - 7.12iT - 23T^{2} \) |
| 29 | \( 1 + 7.06T + 29T^{2} \) |
| 31 | \( 1 + 8.80T + 31T^{2} \) |
| 37 | \( 1 + 6.29T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 - 10.2iT - 43T^{2} \) |
| 47 | \( 1 - 6.63iT - 47T^{2} \) |
| 53 | \( 1 - 1.04iT - 53T^{2} \) |
| 59 | \( 1 - 2.68iT - 59T^{2} \) |
| 61 | \( 1 - 9.03iT - 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 + 3.62iT - 71T^{2} \) |
| 73 | \( 1 + 4.53iT - 73T^{2} \) |
| 79 | \( 1 - 11.4iT - 79T^{2} \) |
| 83 | \( 1 + 14.8T + 83T^{2} \) |
| 89 | \( 1 - 8.05iT - 89T^{2} \) |
| 97 | \( 1 - 9.28T + 97T^{2} \) |
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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
−9.858628183670237342889025638423, −9.211404366409986205728834923616, −8.928751387580657547546147560133, −7.981147709021016665964980255609, −7.41385524349985418803055059718, −5.93516863161967241806488855671, −5.55544259040040038505752391984, −3.67401779583433364324710236546, −2.79457880859739616328176867355, −1.42597761004505405239988893078,
0.19713146324416650542199226354, 1.68110295595931939997190397800, 2.31039886589542112681377578811, 3.59954165614446535243210155405, 5.72976220767870273034048612487, 6.66655702431796467297381392433, 7.21610959405455844780521671507, 7.63506997431111697472674559789, 8.702240356671477457860257460517, 9.182784743704047862122889551311
