| L(s) = 1 | − 2-s − 4-s + 4·7-s + 3·8-s − 2·11-s + 13-s − 4·14-s − 16-s + 2·17-s − 6·19-s + 2·22-s − 6·23-s − 26-s − 4·28-s − 2·29-s − 10·31-s − 5·32-s − 2·34-s + 2·37-s + 6·38-s + 6·41-s − 10·43-s + 2·44-s + 6·46-s + 4·47-s + 9·49-s − 52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.51·7-s + 1.06·8-s − 0.603·11-s + 0.277·13-s − 1.06·14-s − 1/4·16-s + 0.485·17-s − 1.37·19-s + 0.426·22-s − 1.25·23-s − 0.196·26-s − 0.755·28-s − 0.371·29-s − 1.79·31-s − 0.883·32-s − 0.342·34-s + 0.328·37-s + 0.973·38-s + 0.937·41-s − 1.52·43-s + 0.301·44-s + 0.884·46-s + 0.583·47-s + 9/7·49-s − 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.416978267450511008558400223144, −7.82106891920574807374790883803, −7.29036892297292589117564354937, −5.96878859103344980820872258674, −5.24203354891723185421487064499, −4.46467149067514452898640015031, −3.78639204002347269783690843092, −2.18840342844256460053861264523, −1.45787212361509739096947078392, 0,
1.45787212361509739096947078392, 2.18840342844256460053861264523, 3.78639204002347269783690843092, 4.46467149067514452898640015031, 5.24203354891723185421487064499, 5.96878859103344980820872258674, 7.29036892297292589117564354937, 7.82106891920574807374790883803, 8.416978267450511008558400223144
