L(s) = 1 | − 2.49·3-s + (−2.19 − 0.417i)5-s + 3.04i·7-s + 3.24·9-s − 5.87·11-s + 6.00i·13-s + (5.48 + 1.04i)15-s − 0.968·17-s − 1.72·19-s − 7.61i·21-s + (−3.79 + 2.93i)23-s + (4.65 + 1.83i)25-s − 0.611·27-s − 3.89·29-s + 9.01i·31-s + ⋯ |
L(s) = 1 | − 1.44·3-s + (−0.982 − 0.186i)5-s + 1.15i·7-s + 1.08·9-s − 1.77·11-s + 1.66i·13-s + (1.41 + 0.269i)15-s − 0.234·17-s − 0.395·19-s − 1.66i·21-s + (−0.790 + 0.612i)23-s + (0.930 + 0.366i)25-s − 0.117·27-s − 0.723·29-s + 1.61i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0522 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0522 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.07773047811\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.07773047811\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (2.19 + 0.417i)T \) |
| 23 | \( 1 + (3.79 - 2.93i)T \) |
good | 3 | \( 1 + 2.49T + 3T^{2} \) |
| 7 | \( 1 - 3.04iT - 7T^{2} \) |
| 11 | \( 1 + 5.87T + 11T^{2} \) |
| 13 | \( 1 - 6.00iT - 13T^{2} \) |
| 17 | \( 1 + 0.968T + 17T^{2} \) |
| 19 | \( 1 + 1.72T + 19T^{2} \) |
| 29 | \( 1 + 3.89T + 29T^{2} \) |
| 31 | \( 1 - 9.01iT - 31T^{2} \) |
| 37 | \( 1 - 2.15T + 37T^{2} \) |
| 41 | \( 1 - 6.17T + 41T^{2} \) |
| 43 | \( 1 - 9.08iT - 43T^{2} \) |
| 47 | \( 1 + 9.73T + 47T^{2} \) |
| 53 | \( 1 + 9.18T + 53T^{2} \) |
| 59 | \( 1 + 1.17iT - 59T^{2} \) |
| 61 | \( 1 - 0.951iT - 61T^{2} \) |
| 67 | \( 1 + 9.83iT - 67T^{2} \) |
| 71 | \( 1 - 8.99iT - 71T^{2} \) |
| 73 | \( 1 + 8.53iT - 73T^{2} \) |
| 79 | \( 1 + 5.59T + 79T^{2} \) |
| 83 | \( 1 + 5.97iT - 83T^{2} \) |
| 89 | \( 1 + 7.22iT - 89T^{2} \) |
| 97 | \( 1 + 7.51T + 97T^{2} \) |
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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
−9.910319189029532329768416445764, −9.014570903261871086222783752830, −8.212628572437510891562718301169, −7.42178185301083086271332702218, −6.49910635310289200803146436962, −5.79366087503849422610007739726, −4.94722805858277034998300920446, −4.48343477897121511692159230438, −3.07827045400599393820689247418, −1.80644240610183202591737178963,
0.06532892183612302590474994814, 0.58209562456674250556856457191, 2.64129419842986683363158689457, 3.82035045033909276892919543271, 4.64106357371084057966765396862, 5.42736209649192702100577739926, 6.14762643812793846917425826101, 7.18289591901112218422091624184, 7.79359655989232759672509722197, 8.243584935765568905749363242575