| L(s) = 1 | + 2·2-s + 4·4-s − 23·7-s + 8·8-s + 30·11-s + 34·13-s − 46·14-s + 16·16-s + 42·17-s − 139·19-s + 60·22-s − 192·23-s + 68·26-s − 92·28-s + 234·29-s − 55·31-s + 32·32-s + 84·34-s − 191·37-s − 278·38-s + 138·41-s − 53·43-s + 120·44-s − 384·46-s − 366·47-s + 186·49-s + 136·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.24·7-s + 0.353·8-s + 0.822·11-s + 0.725·13-s − 0.878·14-s + 1/4·16-s + 0.599·17-s − 1.67·19-s + 0.581·22-s − 1.74·23-s + 0.512·26-s − 0.620·28-s + 1.49·29-s − 0.318·31-s + 0.176·32-s + 0.423·34-s − 0.848·37-s − 1.18·38-s + 0.525·41-s − 0.187·43-s + 0.411·44-s − 1.23·46-s − 1.13·47-s + 0.542·49-s + 0.362·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 23 T + p^{3} T^{2} \) |
| 11 | \( 1 - 30 T + p^{3} T^{2} \) |
| 13 | \( 1 - 34 T + p^{3} T^{2} \) |
| 17 | \( 1 - 42 T + p^{3} T^{2} \) |
| 19 | \( 1 + 139 T + p^{3} T^{2} \) |
| 23 | \( 1 + 192 T + p^{3} T^{2} \) |
| 29 | \( 1 - 234 T + p^{3} T^{2} \) |
| 31 | \( 1 + 55 T + p^{3} T^{2} \) |
| 37 | \( 1 + 191 T + p^{3} T^{2} \) |
| 41 | \( 1 - 138 T + p^{3} T^{2} \) |
| 43 | \( 1 + 53 T + p^{3} T^{2} \) |
| 47 | \( 1 + 366 T + p^{3} T^{2} \) |
| 53 | \( 1 - 330 T + p^{3} T^{2} \) |
| 59 | \( 1 + 396 T + p^{3} T^{2} \) |
| 61 | \( 1 - 23 T + p^{3} T^{2} \) |
| 67 | \( 1 + 452 T + p^{3} T^{2} \) |
| 71 | \( 1 - 204 T + p^{3} T^{2} \) |
| 73 | \( 1 - 691 T + p^{3} T^{2} \) |
| 79 | \( 1 + 709 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1098 T + p^{3} T^{2} \) |
| 89 | \( 1 + 816 T + p^{3} T^{2} \) |
| 97 | \( 1 + 905 T + p^{3} T^{2} \) |
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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
−8.780700555398891059209052716080, −8.055202283468736060250205471336, −6.74623687761812631476890764817, −6.39711725060808475924125895120, −5.65873074442689504957433667025, −4.29552379696968446996877054306, −3.74316705590031153561713928031, −2.76632949709378067294710156685, −1.53556454670115223389719528227, 0,
1.53556454670115223389719528227, 2.76632949709378067294710156685, 3.74316705590031153561713928031, 4.29552379696968446996877054306, 5.65873074442689504957433667025, 6.39711725060808475924125895120, 6.74623687761812631476890764817, 8.055202283468736060250205471336, 8.780700555398891059209052716080
