| L(s) = 1 | + 2·3-s + 3·7-s + 9-s − 5·11-s − 4·13-s + 3·17-s + 19-s + 6·21-s − 8·23-s − 4·27-s + 2·29-s + 4·31-s − 10·33-s + 10·37-s − 8·39-s + 10·41-s + 43-s + 47-s + 2·49-s + 6·51-s − 4·53-s + 2·57-s − 6·59-s + 13·61-s + 3·63-s − 12·67-s − 16·69-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.13·7-s + 1/3·9-s − 1.50·11-s − 1.10·13-s + 0.727·17-s + 0.229·19-s + 1.30·21-s − 1.66·23-s − 0.769·27-s + 0.371·29-s + 0.718·31-s − 1.74·33-s + 1.64·37-s − 1.28·39-s + 1.56·41-s + 0.152·43-s + 0.145·47-s + 2/7·49-s + 0.840·51-s − 0.549·53-s + 0.264·57-s − 0.781·59-s + 1.66·61-s + 0.377·63-s − 1.46·67-s − 1.92·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.160310002\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.160310002\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−14.84932781819887, −14.57552175633121, −14.20279816166558, −13.69983524403766, −13.07737589136557, −12.58037009185589, −11.89125931266160, −11.48382494019127, −10.74802304625563, −10.12470158471244, −9.782754462510385, −9.144807057781736, −8.361000482049493, −7.928131048936116, −7.742537184339571, −7.256446214216178, −5.995233748116535, −5.693843636737716, −4.727873013329117, −4.526874577523993, −3.565973966753752, −2.725241676942287, −2.443682705193878, −1.752402626409378, −0.6085239891929836,
0.6085239891929836, 1.752402626409378, 2.443682705193878, 2.725241676942287, 3.565973966753752, 4.526874577523993, 4.727873013329117, 5.693843636737716, 5.995233748116535, 7.256446214216178, 7.742537184339571, 7.928131048936116, 8.361000482049493, 9.144807057781736, 9.782754462510385, 10.12470158471244, 10.74802304625563, 11.48382494019127, 11.89125931266160, 12.58037009185589, 13.07737589136557, 13.69983524403766, 14.20279816166558, 14.57552175633121, 14.84932781819887
