| L(s) = 1 | + 3-s + 9-s − 4·11-s − 13-s − 8·17-s + 5·19-s + 4·23-s + 27-s + 9·29-s − 4·31-s − 4·33-s + 3·37-s − 39-s + 5·41-s − 6·43-s + 5·47-s − 7·49-s − 8·51-s + 5·53-s + 5·57-s − 6·59-s + 4·61-s + 3·67-s + 4·69-s + 7·71-s − 4·73-s − 79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 1.20·11-s − 0.277·13-s − 1.94·17-s + 1.14·19-s + 0.834·23-s + 0.192·27-s + 1.67·29-s − 0.718·31-s − 0.696·33-s + 0.493·37-s − 0.160·39-s + 0.780·41-s − 0.914·43-s + 0.729·47-s − 49-s − 1.12·51-s + 0.686·53-s + 0.662·57-s − 0.781·59-s + 0.512·61-s + 0.366·67-s + 0.481·69-s + 0.830·71-s − 0.468·73-s − 0.112·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.094160526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.094160526\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−7.903131603519908289747785665839, −7.17993141418452131960325120897, −6.67617368485630854605762706128, −5.71600405991886094064258589832, −4.87786463283473746888550723725, −4.46324289568589546307923275372, −3.32051839779921621955058449345, −2.69514179703195495755979593252, −2.00064156841136673098677415870, −0.67905688090664365485868193436,
0.67905688090664365485868193436, 2.00064156841136673098677415870, 2.69514179703195495755979593252, 3.32051839779921621955058449345, 4.46324289568589546307923275372, 4.87786463283473746888550723725, 5.71600405991886094064258589832, 6.67617368485630854605762706128, 7.17993141418452131960325120897, 7.903131603519908289747785665839
