| L(s) = 1 | − 3-s + 9-s + 5.65·11-s − 5.65·13-s + 5.65·17-s + 4·19-s − 5.65·23-s − 5·25-s − 27-s + 6·29-s − 8·31-s − 5.65·33-s − 2·37-s + 5.65·39-s − 5.65·41-s − 8·47-s − 5.65·51-s + 2·53-s − 4·57-s + 4·59-s − 5.65·61-s − 11.3·67-s + 5.65·69-s + 5.65·71-s − 11.3·73-s + 5·75-s + 11.3·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.333·9-s + 1.70·11-s − 1.56·13-s + 1.37·17-s + 0.917·19-s − 1.17·23-s − 25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.984·33-s − 0.328·37-s + 0.905·39-s − 0.883·41-s − 1.16·47-s − 0.792·51-s + 0.274·53-s − 0.529·57-s + 0.520·59-s − 0.724·61-s − 1.38·67-s + 0.681·69-s + 0.671·71-s − 1.32·73-s + 0.577·75-s + 1.27·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 - 5.65T + 11T^{2} \) |
| 13 | \( 1 + 5.65T + 13T^{2} \) |
| 17 | \( 1 - 5.65T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 5.65T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 5.65T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 5.65T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 - 5.65T + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−7.38763552863504066914788983011, −6.66180757883093863062879148413, −5.97861324327630514907772698902, −5.34404050852531286507562869394, −4.64992658910754387744825163086, −3.83287944982740122886710933174, −3.20394493316144183179453579989, −1.97285604311821539407399239925, −1.23677812021589812783982408537, 0,
1.23677812021589812783982408537, 1.97285604311821539407399239925, 3.20394493316144183179453579989, 3.83287944982740122886710933174, 4.64992658910754387744825163086, 5.34404050852531286507562869394, 5.97861324327630514907772698902, 6.66180757883093863062879148413, 7.38763552863504066914788983011
