L(s) = 1 | − 2-s + 3-s + 4-s + 0.307·5-s − 6-s − 8-s + 9-s − 0.307·10-s + 1.25·11-s + 12-s − 13-s + 0.307·15-s + 16-s − 2.64·17-s − 18-s − 7.86·19-s + 0.307·20-s − 1.25·22-s − 6.47·23-s − 24-s − 4.90·25-s + 26-s + 27-s + 8.88·29-s − 0.307·30-s + 5.22·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.137·5-s − 0.408·6-s − 0.353·8-s + 0.333·9-s − 0.0973·10-s + 0.379·11-s + 0.288·12-s − 0.277·13-s + 0.0794·15-s + 0.250·16-s − 0.640·17-s − 0.235·18-s − 1.80·19-s + 0.0688·20-s − 0.268·22-s − 1.35·23-s − 0.204·24-s − 0.981·25-s + 0.196·26-s + 0.192·27-s + 1.64·29-s − 0.0561·30-s + 0.937·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.581379613\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.581379613\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 - 0.307T + 5T^{2} \) |
| 11 | \( 1 - 1.25T + 11T^{2} \) |
| 17 | \( 1 + 2.64T + 17T^{2} \) |
| 19 | \( 1 + 7.86T + 19T^{2} \) |
| 23 | \( 1 + 6.47T + 23T^{2} \) |
| 29 | \( 1 - 8.88T + 29T^{2} \) |
| 31 | \( 1 - 5.22T + 31T^{2} \) |
| 37 | \( 1 - 8.44T + 37T^{2} \) |
| 41 | \( 1 - 0.192T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 - 9.74T + 47T^{2} \) |
| 53 | \( 1 - 7.74T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 - 9.38T + 61T^{2} \) |
| 67 | \( 1 - 9.62T + 67T^{2} \) |
| 71 | \( 1 + 1.56T + 71T^{2} \) |
| 73 | \( 1 - 6.83T + 73T^{2} \) |
| 79 | \( 1 + 16.0T + 79T^{2} \) |
| 83 | \( 1 + 1.04T + 83T^{2} \) |
| 89 | \( 1 - 1.24T + 89T^{2} \) |
| 97 | \( 1 + 15.7T + 97T^{2} \) |
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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
−8.461809281148835459441911863134, −8.033091449243769650744377412807, −7.10500887320851550276295576819, −6.40699746058824495024569726949, −5.80166163180785528055370342773, −4.36899747940698541643325529703, −4.00729624915548842563214239692, −2.49907063725717392323958972776, −2.20470179890082154909902585444, −0.77275235096140794040527753996,
0.77275235096140794040527753996, 2.20470179890082154909902585444, 2.49907063725717392323958972776, 4.00729624915548842563214239692, 4.36899747940698541643325529703, 5.80166163180785528055370342773, 6.40699746058824495024569726949, 7.10500887320851550276295576819, 8.033091449243769650744377412807, 8.461809281148835459441911863134