| L(s) = 1 | − 1.96i·2-s + 0.187i·3-s − 1.84·4-s + (1.81 − 1.31i)5-s + 0.368·6-s − 0.677i·7-s − 0.297i·8-s + 2.96·9-s + (−2.57 − 3.55i)10-s + 2.84·11-s − 0.346i·12-s + 4.76i·13-s − 1.32·14-s + (0.246 + 0.339i)15-s − 4.27·16-s − 5.18i·17-s + ⋯ |
| L(s) = 1 | − 1.38i·2-s + 0.108i·3-s − 0.924·4-s + (0.809 − 0.586i)5-s + 0.150·6-s − 0.255i·7-s − 0.105i·8-s + 0.988·9-s + (−0.813 − 1.12i)10-s + 0.856·11-s − 0.100i·12-s + 1.32i·13-s − 0.354·14-s + (0.0635 + 0.0877i)15-s − 1.06·16-s − 1.25i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.809 + 0.586i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.809 + 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.314400851\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.314400851\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.81 + 1.31i)T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 1.96iT - 2T^{2} \) |
| 3 | \( 1 - 0.187iT - 3T^{2} \) |
| 7 | \( 1 + 0.677iT - 7T^{2} \) |
| 11 | \( 1 - 2.84T + 11T^{2} \) |
| 13 | \( 1 - 4.76iT - 13T^{2} \) |
| 17 | \( 1 + 5.18iT - 17T^{2} \) |
| 23 | \( 1 - 1.05iT - 23T^{2} \) |
| 29 | \( 1 - 1.42T + 29T^{2} \) |
| 31 | \( 1 - 0.271T + 31T^{2} \) |
| 37 | \( 1 + 0.603iT - 37T^{2} \) |
| 41 | \( 1 - 6.73T + 41T^{2} \) |
| 43 | \( 1 + 5.62iT - 43T^{2} \) |
| 47 | \( 1 + 7.89iT - 47T^{2} \) |
| 53 | \( 1 - 6.88iT - 53T^{2} \) |
| 59 | \( 1 - 10.1T + 59T^{2} \) |
| 61 | \( 1 + 7.47T + 61T^{2} \) |
| 67 | \( 1 + 4.11iT - 67T^{2} \) |
| 71 | \( 1 + 7.60T + 71T^{2} \) |
| 73 | \( 1 - 16.1iT - 73T^{2} \) |
| 79 | \( 1 + 14.8T + 79T^{2} \) |
| 83 | \( 1 - 14.1iT - 83T^{2} \) |
| 89 | \( 1 + 8.23T + 89T^{2} \) |
| 97 | \( 1 - 5.72iT - 97T^{2} \) |
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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
−9.353420746686673632949163018639, −8.694187355870790057314867149505, −7.12583049199463592497887757981, −6.76220563534895947000162856889, −5.48309519155744740904269123908, −4.33603236095144244384621473134, −4.08431229530370084974290567964, −2.67042673365217023372126308675, −1.73934987732277750023322222519, −0.973700512651630692659809086545,
1.45998221197743618541705695833, 2.69429770103711980270663831776, 3.98330348744041052279722570365, 5.00266221820871992974785388729, 6.05608442051752650676538753100, 6.19995085481621794835693758970, 7.17113481868016877491050028534, 7.77343925807513418919618717467, 8.624037554659033308615750683398, 9.398351123772485219236114103810
