| L(s) = 1 | + 1.84·3-s + 0.871·5-s − 3.24·7-s + 0.418·9-s + 2.03·11-s − 0.211·13-s + 1.61·15-s + 6.97·17-s + 19-s − 5.99·21-s + 2.72·23-s − 4.24·25-s − 4.77·27-s − 2.83·29-s − 6.88·31-s + 3.76·33-s − 2.82·35-s + 6.72·37-s − 0.390·39-s − 3.53·41-s + 10.4·43-s + 0.364·45-s + 7.54·47-s + 3.51·49-s + 12.8·51-s + 12.8·53-s + 1.77·55-s + ⋯ |
| L(s) = 1 | + 1.06·3-s + 0.389·5-s − 1.22·7-s + 0.139·9-s + 0.614·11-s − 0.0585·13-s + 0.416·15-s + 1.69·17-s + 0.229·19-s − 1.30·21-s + 0.568·23-s − 0.848·25-s − 0.918·27-s − 0.525·29-s − 1.23·31-s + 0.655·33-s − 0.477·35-s + 1.10·37-s − 0.0624·39-s − 0.552·41-s + 1.59·43-s + 0.0543·45-s + 1.10·47-s + 0.501·49-s + 1.80·51-s + 1.76·53-s + 0.239·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.851541838\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.851541838\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 3 | \( 1 - 1.84T + 3T^{2} \) |
| 5 | \( 1 - 0.871T + 5T^{2} \) |
| 7 | \( 1 + 3.24T + 7T^{2} \) |
| 11 | \( 1 - 2.03T + 11T^{2} \) |
| 13 | \( 1 + 0.211T + 13T^{2} \) |
| 17 | \( 1 - 6.97T + 17T^{2} \) |
| 23 | \( 1 - 2.72T + 23T^{2} \) |
| 29 | \( 1 + 2.83T + 29T^{2} \) |
| 31 | \( 1 + 6.88T + 31T^{2} \) |
| 37 | \( 1 - 6.72T + 37T^{2} \) |
| 41 | \( 1 + 3.53T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 - 7.54T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 - 6.09T + 59T^{2} \) |
| 61 | \( 1 - 9.61T + 61T^{2} \) |
| 67 | \( 1 + 1.50T + 67T^{2} \) |
| 71 | \( 1 - 0.683T + 71T^{2} \) |
| 73 | \( 1 - 3.52T + 73T^{2} \) |
| 83 | \( 1 + 4.54T + 83T^{2} \) |
| 89 | \( 1 + 7.61T + 89T^{2} \) |
| 97 | \( 1 - 18.9T + 97T^{2} \) |
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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
−8.040583776391632496386780132039, −7.44655664426778224236640093213, −6.80957244433462450911434821527, −5.71036277881547881326381719596, −5.61110605766490419761881332728, −4.01732797103656726825736597803, −3.57945973071748654081100309015, −2.85532983393382127327290895383, −2.07633573692056327240209788969, −0.841857274371715811543115526007,
0.841857274371715811543115526007, 2.07633573692056327240209788969, 2.85532983393382127327290895383, 3.57945973071748654081100309015, 4.01732797103656726825736597803, 5.61110605766490419761881332728, 5.71036277881547881326381719596, 6.80957244433462450911434821527, 7.44655664426778224236640093213, 8.040583776391632496386780132039
