L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 0.585·11-s + 12-s + 3·13-s + 16-s + 0.171·17-s − 18-s − 1.41·19-s − 0.585·22-s + 2.41·23-s − 24-s − 3·26-s + 27-s + 29-s + 0.414·31-s − 32-s + 0.585·33-s − 0.171·34-s + 36-s − 7.07·37-s + 1.41·38-s + 3·39-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s − 0.353·8-s + 0.333·9-s + 0.176·11-s + 0.288·12-s + 0.832·13-s + 0.250·16-s + 0.0416·17-s − 0.235·18-s − 0.324·19-s − 0.124·22-s + 0.503·23-s − 0.204·24-s − 0.588·26-s + 0.192·27-s + 0.185·29-s + 0.0743·31-s − 0.176·32-s + 0.101·33-s − 0.0294·34-s + 0.166·36-s − 1.16·37-s + 0.229·38-s + 0.480·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.949140034\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.949140034\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 0.585T + 11T^{2} \) |
| 13 | \( 1 - 3T + 13T^{2} \) |
| 17 | \( 1 - 0.171T + 17T^{2} \) |
| 19 | \( 1 + 1.41T + 19T^{2} \) |
| 23 | \( 1 - 2.41T + 23T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 - 0.414T + 31T^{2} \) |
| 37 | \( 1 + 7.07T + 37T^{2} \) |
| 41 | \( 1 + 0.171T + 41T^{2} \) |
| 43 | \( 1 - 4.41T + 43T^{2} \) |
| 47 | \( 1 - 9.65T + 47T^{2} \) |
| 53 | \( 1 - 1.34T + 53T^{2} \) |
| 59 | \( 1 + 1.24T + 59T^{2} \) |
| 61 | \( 1 + 5.82T + 61T^{2} \) |
| 67 | \( 1 - 7.31T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 - 0.928T + 73T^{2} \) |
| 79 | \( 1 + 13.0T + 79T^{2} \) |
| 83 | \( 1 - 14.0T + 83T^{2} \) |
| 89 | \( 1 - 4T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−8.041555851186913625288322312294, −7.31193330571970690133007942501, −6.69733594249245796766209872567, −5.99100082271201054162744445338, −5.14136987488770864214114403276, −4.13248206154617541701432238752, −3.45478792375148122724305579183, −2.59448175784176780295541620585, −1.73016443050541720897329165760, −0.78379502172290067623818240660,
0.78379502172290067623818240660, 1.73016443050541720897329165760, 2.59448175784176780295541620585, 3.45478792375148122724305579183, 4.13248206154617541701432238752, 5.14136987488770864214114403276, 5.99100082271201054162744445338, 6.69733594249245796766209872567, 7.31193330571970690133007942501, 8.041555851186913625288322312294