L(s) = 1 | − i·3-s + 3.63·5-s + (0.122 − 2.64i)7-s − 9-s − 1.30i·11-s − 0.939i·13-s − 3.63i·15-s + 5.95·17-s − 7.57·19-s + (−2.64 − 0.122i)21-s + (0.194 − 4.79i)23-s + 8.23·25-s + i·27-s − 1.09·29-s − 4.06i·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + 1.62·5-s + (0.0462 − 0.998i)7-s − 0.333·9-s − 0.392i·11-s − 0.260i·13-s − 0.939i·15-s + 1.44·17-s − 1.73·19-s + (−0.576 − 0.0266i)21-s + (0.0405 − 0.999i)23-s + 1.64·25-s + 0.192i·27-s − 0.202·29-s − 0.730i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0867 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0867 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.263404286\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.263404286\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 + (-0.122 + 2.64i)T \) |
| 23 | \( 1 + (-0.194 + 4.79i)T \) |
good | 5 | \( 1 - 3.63T + 5T^{2} \) |
| 11 | \( 1 + 1.30iT - 11T^{2} \) |
| 13 | \( 1 + 0.939iT - 13T^{2} \) |
| 17 | \( 1 - 5.95T + 17T^{2} \) |
| 19 | \( 1 + 7.57T + 19T^{2} \) |
| 29 | \( 1 + 1.09T + 29T^{2} \) |
| 31 | \( 1 + 4.06iT - 31T^{2} \) |
| 37 | \( 1 - 7.18iT - 37T^{2} \) |
| 41 | \( 1 + 7.59iT - 41T^{2} \) |
| 43 | \( 1 + 0.163iT - 43T^{2} \) |
| 47 | \( 1 + 3.48iT - 47T^{2} \) |
| 53 | \( 1 - 4.81iT - 53T^{2} \) |
| 59 | \( 1 - 9.59iT - 59T^{2} \) |
| 61 | \( 1 - 5.67T + 61T^{2} \) |
| 67 | \( 1 - 9.71iT - 67T^{2} \) |
| 71 | \( 1 - 9.72T + 71T^{2} \) |
| 73 | \( 1 + 11.9iT - 73T^{2} \) |
| 79 | \( 1 - 0.0817iT - 79T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 - 9.86T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−8.927582003415043658939017112604, −8.247446442378931711696729501482, −7.33190259649989812447601729429, −6.48067179760207913036507101919, −5.97278897346885199279701396175, −5.13603757396530037772791750495, −4.00409146589173884391799877907, −2.79408049993399035168873406647, −1.86427585582797896060582871578, −0.814382020260333956356545536440,
1.65718548684894106829497953036, 2.39489785611760285734287248804, 3.46403454724409688458908537777, 4.74938349121820497517650574234, 5.48656741962963844866886897356, 5.99989113984602247083800709856, 6.80114840471886053410567469799, 8.071205629215362589745327145014, 8.821951852632767667506210820118, 9.633255154832486219823625566037