| L(s) = 1 | + 2·3-s + 7-s + 9-s + 5·11-s + 8·17-s − 2·19-s + 2·21-s − 7·23-s − 4·27-s − 3·29-s + 4·31-s + 10·33-s − 37-s − 2·41-s + 3·43-s + 6·47-s + 49-s + 16·51-s + 10·53-s − 4·57-s − 4·59-s − 6·61-s + 63-s + 13·67-s − 14·69-s + 5·71-s + 6·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.377·7-s + 1/3·9-s + 1.50·11-s + 1.94·17-s − 0.458·19-s + 0.436·21-s − 1.45·23-s − 0.769·27-s − 0.557·29-s + 0.718·31-s + 1.74·33-s − 0.164·37-s − 0.312·41-s + 0.457·43-s + 0.875·47-s + 1/7·49-s + 2.24·51-s + 1.37·53-s − 0.529·57-s − 0.520·59-s − 0.768·61-s + 0.125·63-s + 1.58·67-s − 1.68·69-s + 0.593·71-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.814561857\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.814561857\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−9.513870706542333340648205265315, −8.684803347503568812792785869548, −8.069078370838473547557932517577, −7.39070327434938193188335055166, −6.30047245941881715667629546480, −5.45788503200219529211336521976, −4.06748454385499559567921683687, −3.59222171566378595092146482354, −2.39874083819611908400458517412, −1.31531657098182471156833900193,
1.31531657098182471156833900193, 2.39874083819611908400458517412, 3.59222171566378595092146482354, 4.06748454385499559567921683687, 5.45788503200219529211336521976, 6.30047245941881715667629546480, 7.39070327434938193188335055166, 8.069078370838473547557932517577, 8.684803347503568812792785869548, 9.513870706542333340648205265315
