| L(s) = 1 | − 2·2-s + 4·4-s − 8·8-s + 10·13-s + 16·16-s − 16·17-s − 20·26-s + 40·29-s − 32·32-s + 32·34-s + 70·37-s − 80·41-s + 49·49-s + 40·52-s + 56·53-s − 80·58-s − 22·61-s + 64·64-s − 64·68-s − 110·73-s − 140·74-s + 160·82-s + 160·89-s + 130·97-s − 98·98-s + 40·101-s − 80·104-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s − 8-s + 0.769·13-s + 16-s − 0.941·17-s − 0.769·26-s + 1.37·29-s − 32-s + 0.941·34-s + 1.89·37-s − 1.95·41-s + 49-s + 0.769·52-s + 1.05·53-s − 1.37·58-s − 0.360·61-s + 64-s − 0.941·68-s − 1.50·73-s − 1.89·74-s + 1.95·82-s + 1.79·89-s + 1.34·97-s − 98-s + 0.396·101-s − 0.769·104-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.155969460\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.155969460\) |
| \(L(2)\) |
|
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 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( 1 - 10 T + p^{2} T^{2} \) |
| 17 | \( 1 + 16 T + p^{2} T^{2} \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( 1 - 40 T + p^{2} T^{2} \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( 1 - 70 T + p^{2} T^{2} \) |
| 41 | \( 1 + 80 T + p^{2} T^{2} \) |
| 43 | \( ( 1 - p T )( 1 + p T ) \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( 1 - 56 T + p^{2} T^{2} \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 + 22 T + p^{2} T^{2} \) |
| 67 | \( ( 1 - p T )( 1 + p T ) \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 + 110 T + p^{2} T^{2} \) |
| 79 | \( ( 1 - p T )( 1 + p T ) \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( 1 - 160 T + p^{2} T^{2} \) |
| 97 | \( 1 - 130 T + p^{2} T^{2} \) |
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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
−9.952825395670855120177679459651, −8.898391029191983733334903491397, −8.472281502197646096143753675782, −7.47595882023977793530150043357, −6.59491683867461329811696907042, −5.89916341704319828301125561453, −4.53986602554168401189444141480, −3.24375705805022196197764303676, −2.11111853341150925837743673903, −0.796076931157797475650435567915,
0.796076931157797475650435567915, 2.11111853341150925837743673903, 3.24375705805022196197764303676, 4.53986602554168401189444141480, 5.89916341704319828301125561453, 6.59491683867461329811696907042, 7.47595882023977793530150043357, 8.472281502197646096143753675782, 8.898391029191983733334903491397, 9.952825395670855120177679459651
