| L(s) = 1 | − 2·3-s + 7-s + 9-s − 3·11-s − 4·13-s + 3·17-s − 19-s − 2·21-s + 4·27-s − 6·29-s − 4·31-s + 6·33-s + 2·37-s + 8·39-s − 6·41-s − 43-s + 3·47-s − 6·49-s − 6·51-s + 12·53-s + 2·57-s + 6·59-s + 61-s + 63-s − 4·67-s + 6·71-s + 7·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.377·7-s + 1/3·9-s − 0.904·11-s − 1.10·13-s + 0.727·17-s − 0.229·19-s − 0.436·21-s + 0.769·27-s − 1.11·29-s − 0.718·31-s + 1.04·33-s + 0.328·37-s + 1.28·39-s − 0.937·41-s − 0.152·43-s + 0.437·47-s − 6/7·49-s − 0.840·51-s + 1.64·53-s + 0.264·57-s + 0.781·59-s + 0.128·61-s + 0.125·63-s − 0.488·67-s + 0.712·71-s + 0.819·73-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 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
−15.32409483829961, −14.79142324254747, −14.58993995285089, −13.64902871676154, −13.25683502112021, −12.49047695775711, −12.22547896350950, −11.66574073910036, −11.10847964185335, −10.65113291626201, −10.15921211903439, −9.595964201447504, −8.943908841408024, −8.091387705078546, −7.758138400884024, −7.039391845140152, −6.564694339677248, −5.659807020461848, −5.339442291435174, −4.984438268124573, −4.167238298211070, −3.374348780199338, −2.516294310175773, −1.864996398388929, −0.7674398829270769, 0,
0.7674398829270769, 1.864996398388929, 2.516294310175773, 3.374348780199338, 4.167238298211070, 4.984438268124573, 5.339442291435174, 5.659807020461848, 6.564694339677248, 7.039391845140152, 7.758138400884024, 8.091387705078546, 8.943908841408024, 9.595964201447504, 10.15921211903439, 10.65113291626201, 11.10847964185335, 11.66574073910036, 12.22547896350950, 12.49047695775711, 13.25683502112021, 13.64902871676154, 14.58993995285089, 14.79142324254747, 15.32409483829961
