| L(s) = 1 | + 3-s − 2·4-s − 5-s + 2·7-s + 9-s − 5·11-s − 2·12-s − 13-s − 15-s + 4·16-s − 4·17-s + 6·19-s + 2·20-s + 2·21-s + 8·23-s + 25-s + 27-s − 4·28-s + 2·29-s + 8·31-s − 5·33-s − 2·35-s − 2·36-s − 2·37-s − 39-s − 2·43-s + 10·44-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 0.447·5-s + 0.755·7-s + 1/3·9-s − 1.50·11-s − 0.577·12-s − 0.277·13-s − 0.258·15-s + 16-s − 0.970·17-s + 1.37·19-s + 0.447·20-s + 0.436·21-s + 1.66·23-s + 1/5·25-s + 0.192·27-s − 0.755·28-s + 0.371·29-s + 1.43·31-s − 0.870·33-s − 0.338·35-s − 1/3·36-s − 0.328·37-s − 0.160·39-s − 0.304·43-s + 1.50·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.246940528\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.246940528\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| 113 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−13.56893367767878, −13.37283487235685, −13.01129556641856, −12.39994744774150, −11.72598549424488, −11.42432914805116, −10.67991666558147, −10.22425188267284, −9.882541423636981, −9.116374108495048, −8.689056105459410, −8.405133254459811, −7.719843681021612, −7.490093201906313, −6.896519363456196, −6.049035956060765, −5.219406825432178, −4.922774408701186, −4.663249572529054, −3.893623159225519, −3.137254083553421, −2.828161831874441, −2.044915385720546, −1.087466256553313, −0.5331242899698088,
0.5331242899698088, 1.087466256553313, 2.044915385720546, 2.828161831874441, 3.137254083553421, 3.893623159225519, 4.663249572529054, 4.922774408701186, 5.219406825432178, 6.049035956060765, 6.896519363456196, 7.490093201906313, 7.719843681021612, 8.405133254459811, 8.689056105459410, 9.116374108495048, 9.882541423636981, 10.22425188267284, 10.67991666558147, 11.42432914805116, 11.72598549424488, 12.39994744774150, 13.01129556641856, 13.37283487235685, 13.56893367767878
