L(s) = 1 | + 2·2-s + 4·4-s − 26·7-s + 8·8-s + 28·11-s − 12·13-s − 52·14-s + 16·16-s − 64·17-s − 60·19-s + 56·22-s − 58·23-s − 24·26-s − 104·28-s − 90·29-s − 128·31-s + 32·32-s − 128·34-s − 236·37-s − 120·38-s − 242·41-s − 362·43-s + 112·44-s − 116·46-s + 226·47-s + 333·49-s − 48·52-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.40·7-s + 0.353·8-s + 0.767·11-s − 0.256·13-s − 0.992·14-s + 1/4·16-s − 0.913·17-s − 0.724·19-s + 0.542·22-s − 0.525·23-s − 0.181·26-s − 0.701·28-s − 0.576·29-s − 0.741·31-s + 0.176·32-s − 0.645·34-s − 1.04·37-s − 0.512·38-s − 0.921·41-s − 1.28·43-s + 0.383·44-s − 0.371·46-s + 0.701·47-s + 0.970·49-s − 0.128·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 26 T + p^{3} T^{2} \) |
| 11 | \( 1 - 28 T + p^{3} T^{2} \) |
| 13 | \( 1 + 12 T + p^{3} T^{2} \) |
| 17 | \( 1 + 64 T + p^{3} T^{2} \) |
| 19 | \( 1 + 60 T + p^{3} T^{2} \) |
| 23 | \( 1 + 58 T + p^{3} T^{2} \) |
| 29 | \( 1 + 90 T + p^{3} T^{2} \) |
| 31 | \( 1 + 128 T + p^{3} T^{2} \) |
| 37 | \( 1 + 236 T + p^{3} T^{2} \) |
| 41 | \( 1 + 242 T + p^{3} T^{2} \) |
| 43 | \( 1 + 362 T + p^{3} T^{2} \) |
| 47 | \( 1 - 226 T + p^{3} T^{2} \) |
| 53 | \( 1 + 108 T + p^{3} T^{2} \) |
| 59 | \( 1 - 20 T + p^{3} T^{2} \) |
| 61 | \( 1 - 542 T + p^{3} T^{2} \) |
| 67 | \( 1 - 434 T + p^{3} T^{2} \) |
| 71 | \( 1 - 1128 T + p^{3} T^{2} \) |
| 73 | \( 1 + 632 T + p^{3} T^{2} \) |
| 79 | \( 1 + 720 T + p^{3} T^{2} \) |
| 83 | \( 1 + 478 T + p^{3} T^{2} \) |
| 89 | \( 1 - 490 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1456 T + p^{3} T^{2} \) |
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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
−10.24705929574650852945409835618, −9.427606068435365467927169499195, −8.473027275732949666847633909369, −6.97237505725871307599391348466, −6.55155528932095901846697571472, −5.48633302056199884963434598773, −4.16400255317564643962006062450, −3.34639866936807531453688209814, −2.02021459137207957754950848163, 0,
2.02021459137207957754950848163, 3.34639866936807531453688209814, 4.16400255317564643962006062450, 5.48633302056199884963434598773, 6.55155528932095901846697571472, 6.97237505725871307599391348466, 8.473027275732949666847633909369, 9.427606068435365467927169499195, 10.24705929574650852945409835618