| L(s) = 1 | − 2.28·3-s + 2.82·7-s + 2.23·9-s + 5.23·11-s − 4.03·13-s + 1.08·17-s + 19-s − 6.47·21-s − 7.40·23-s + 1.74·27-s + 4.47·29-s + 4·31-s − 11.9·33-s − 6.86·37-s + 9.23·39-s − 6·41-s + 8.48·43-s + 8.48·47-s + 1.00·49-s − 2.47·51-s + 6.86·53-s − 2.28·57-s + 10.4·59-s + 1.70·61-s + 6.32·63-s + 1.62·67-s + 16.9·69-s + ⋯ |
| L(s) = 1 | − 1.32·3-s + 1.06·7-s + 0.745·9-s + 1.57·11-s − 1.11·13-s + 0.262·17-s + 0.229·19-s − 1.41·21-s − 1.54·23-s + 0.336·27-s + 0.830·29-s + 0.718·31-s − 2.08·33-s − 1.12·37-s + 1.47·39-s − 0.937·41-s + 1.29·43-s + 1.23·47-s + 0.142·49-s − 0.346·51-s + 0.942·53-s − 0.303·57-s + 1.36·59-s + 0.218·61-s + 0.796·63-s + 0.197·67-s + 2.03·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.424006265\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.424006265\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2.28T + 3T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 - 5.23T + 11T^{2} \) |
| 13 | \( 1 + 4.03T + 13T^{2} \) |
| 17 | \( 1 - 1.08T + 17T^{2} \) |
| 23 | \( 1 + 7.40T + 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 6.86T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 8.48T + 43T^{2} \) |
| 47 | \( 1 - 8.48T + 47T^{2} \) |
| 53 | \( 1 - 6.86T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 - 1.70T + 61T^{2} \) |
| 67 | \( 1 - 1.62T + 67T^{2} \) |
| 71 | \( 1 - 1.52T + 71T^{2} \) |
| 73 | \( 1 - 13.7T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 + 5.24T + 83T^{2} \) |
| 89 | \( 1 + 14.9T + 89T^{2} \) |
| 97 | \( 1 + 4.44T + 97T^{2} \) |
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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
−7.86347327828652869039244175337, −6.87130472627400063102985243831, −6.61605317474960946978046256378, −5.59855664757915759636789556990, −5.25402355038397061775629536603, −4.38110821284434895366922806772, −3.90829183231423985512932818805, −2.51037852448018197663888900460, −1.54773244320879849943648833069, −0.68189629917302463066420751895,
0.68189629917302463066420751895, 1.54773244320879849943648833069, 2.51037852448018197663888900460, 3.90829183231423985512932818805, 4.38110821284434895366922806772, 5.25402355038397061775629536603, 5.59855664757915759636789556990, 6.61605317474960946978046256378, 6.87130472627400063102985243831, 7.86347327828652869039244175337
