| L(s) = 1 | − 3·3-s + 10·5-s + 9·9-s − 52·11-s + 10·13-s − 30·15-s + 54·17-s + 52·19-s + 48·23-s − 25·25-s − 27·27-s − 186·29-s − 224·31-s + 156·33-s + 94·37-s − 30·39-s + 478·41-s − 316·43-s + 90·45-s − 256·47-s − 162·51-s − 66·53-s − 520·55-s − 156·57-s − 420·59-s − 342·61-s + 100·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1/3·9-s − 1.42·11-s + 0.213·13-s − 0.516·15-s + 0.770·17-s + 0.627·19-s + 0.435·23-s − 1/5·25-s − 0.192·27-s − 1.19·29-s − 1.29·31-s + 0.822·33-s + 0.417·37-s − 0.123·39-s + 1.82·41-s − 1.12·43-s + 0.298·45-s − 0.794·47-s − 0.444·51-s − 0.171·53-s − 1.27·55-s − 0.362·57-s − 0.926·59-s − 0.717·61-s + 0.190·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 11 | \( 1 + 52 T + p^{3} T^{2} \) |
| 13 | \( 1 - 10 T + p^{3} T^{2} \) |
| 17 | \( 1 - 54 T + p^{3} T^{2} \) |
| 19 | \( 1 - 52 T + p^{3} T^{2} \) |
| 23 | \( 1 - 48 T + p^{3} T^{2} \) |
| 29 | \( 1 + 186 T + p^{3} T^{2} \) |
| 31 | \( 1 + 224 T + p^{3} T^{2} \) |
| 37 | \( 1 - 94 T + p^{3} T^{2} \) |
| 41 | \( 1 - 478 T + p^{3} T^{2} \) |
| 43 | \( 1 + 316 T + p^{3} T^{2} \) |
| 47 | \( 1 + 256 T + p^{3} T^{2} \) |
| 53 | \( 1 + 66 T + p^{3} T^{2} \) |
| 59 | \( 1 + 420 T + p^{3} T^{2} \) |
| 61 | \( 1 + 342 T + p^{3} T^{2} \) |
| 67 | \( 1 - 668 T + p^{3} T^{2} \) |
| 71 | \( 1 + 272 T + p^{3} T^{2} \) |
| 73 | \( 1 - 86 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1360 T + p^{3} T^{2} \) |
| 83 | \( 1 + 188 T + p^{3} T^{2} \) |
| 89 | \( 1 - 366 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1554 T + p^{3} T^{2} \) |
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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
−9.301028175307048639837826058601, −7.964729100558059615306700382729, −7.41904445149583084739713295766, −6.26152210922181248015222742217, −5.51153440931212913545881196080, −5.05563085026384984109970815977, −3.63899531286361187765413820523, −2.49733117017101402385610157020, −1.39234914392480450159270336031, 0,
1.39234914392480450159270336031, 2.49733117017101402385610157020, 3.63899531286361187765413820523, 5.05563085026384984109970815977, 5.51153440931212913545881196080, 6.26152210922181248015222742217, 7.41904445149583084739713295766, 7.964729100558059615306700382729, 9.301028175307048639837826058601
