| L(s) = 1 | − 2-s − 3.25·3-s + 4-s + 3.25·6-s − 1.44·7-s − 8-s + 7.62·9-s − 2.03·11-s − 3.25·12-s − 0.557·13-s + 1.44·14-s + 16-s + 3.91·17-s − 7.62·18-s − 6.73·19-s + 4.70·21-s + 2.03·22-s − 23-s + 3.25·24-s + 0.557·26-s − 15.0·27-s − 1.44·28-s − 9.69·29-s − 3.07·31-s − 32-s + 6.62·33-s − 3.91·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.88·3-s + 0.5·4-s + 1.33·6-s − 0.545·7-s − 0.353·8-s + 2.54·9-s − 0.612·11-s − 0.940·12-s − 0.154·13-s + 0.385·14-s + 0.250·16-s + 0.950·17-s − 1.79·18-s − 1.54·19-s + 1.02·21-s + 0.433·22-s − 0.208·23-s + 0.665·24-s + 0.109·26-s − 2.89·27-s − 0.272·28-s − 1.80·29-s − 0.552·31-s − 0.176·32-s + 1.15·33-s − 0.672·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3511649334\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3511649334\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 3.25T + 3T^{2} \) |
| 7 | \( 1 + 1.44T + 7T^{2} \) |
| 11 | \( 1 + 2.03T + 11T^{2} \) |
| 13 | \( 1 + 0.557T + 13T^{2} \) |
| 17 | \( 1 - 3.91T + 17T^{2} \) |
| 19 | \( 1 + 6.73T + 19T^{2} \) |
| 29 | \( 1 + 9.69T + 29T^{2} \) |
| 31 | \( 1 + 3.07T + 31T^{2} \) |
| 37 | \( 1 - 3.65T + 37T^{2} \) |
| 41 | \( 1 - 7.03T + 41T^{2} \) |
| 43 | \( 1 - 5.06T + 43T^{2} \) |
| 47 | \( 1 + 0.659T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 - 1.73T + 61T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 - 9.26T + 73T^{2} \) |
| 79 | \( 1 - 5.92T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 - 5.25T + 89T^{2} \) |
| 97 | \( 1 - 0.0813T + 97T^{2} \) |
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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
−9.910347316405260926220814607573, −9.335829084044317779271287710236, −7.929507893686179080040443064409, −7.26049138930990715708334617566, −6.30008608411438345460778776894, −5.82893848064003505912976947145, −4.89862752587522213910460608873, −3.74880591123635460662934626289, −2.01536568514345823129837498352, −0.52858867159432612116623557094,
0.52858867159432612116623557094, 2.01536568514345823129837498352, 3.74880591123635460662934626289, 4.89862752587522213910460608873, 5.82893848064003505912976947145, 6.30008608411438345460778776894, 7.26049138930990715708334617566, 7.929507893686179080040443064409, 9.335829084044317779271287710236, 9.910347316405260926220814607573
