| L(s) = 1 | − 3-s + 9-s − 11-s + 4·13-s + 6·17-s − 7·19-s + 23-s − 27-s − 6·29-s − 4·31-s + 33-s + 2·37-s − 4·39-s − 9·41-s − 2·43-s + 7·47-s − 6·51-s − 5·53-s + 7·57-s − 7·59-s + 7·61-s + 2·67-s − 69-s + 4·71-s − 10·73-s − 6·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.301·11-s + 1.10·13-s + 1.45·17-s − 1.60·19-s + 0.208·23-s − 0.192·27-s − 1.11·29-s − 0.718·31-s + 0.174·33-s + 0.328·37-s − 0.640·39-s − 1.40·41-s − 0.304·43-s + 1.02·47-s − 0.840·51-s − 0.686·53-s + 0.927·57-s − 0.911·59-s + 0.896·61-s + 0.244·67-s − 0.120·69-s + 0.474·71-s − 1.17·73-s − 0.675·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.184371218\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.184371218\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−12.66607600158088, −12.04098921304145, −11.73695227901209, −11.10172423744492, −10.79723920328481, −10.44107981520189, −9.957697216981817, −9.399715362674537, −8.929858763708406, −8.409051639173460, −8.004712676017556, −7.496551622492678, −6.983683105198493, −6.423652811111166, −6.047511921865927, −5.526130735740888, −5.206259453585692, −4.517664591885914, −3.951856884641765, −3.558743907023993, −3.018949442200952, −2.199021625728533, −1.657506566569528, −1.123567430898201, −0.3130307407169617,
0.3130307407169617, 1.123567430898201, 1.657506566569528, 2.199021625728533, 3.018949442200952, 3.558743907023993, 3.951856884641765, 4.517664591885914, 5.206259453585692, 5.526130735740888, 6.047511921865927, 6.423652811111166, 6.983683105198493, 7.496551622492678, 8.004712676017556, 8.409051639173460, 8.929858763708406, 9.399715362674537, 9.957697216981817, 10.44107981520189, 10.79723920328481, 11.10172423744492, 11.73695227901209, 12.04098921304145, 12.66607600158088
