| L(s) = 1 | − 3-s + 9-s − 2·11-s − 13-s + 4·17-s − 5·19-s − 6·23-s − 27-s + 8·29-s + 2·33-s + 9·37-s + 39-s − 6·41-s − 4·43-s + 4·47-s − 4·51-s + 5·57-s + 4·59-s + 11·61-s − 5·67-s + 6·69-s − 8·71-s − 73-s − 5·79-s + 81-s − 4·83-s − 8·87-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.603·11-s − 0.277·13-s + 0.970·17-s − 1.14·19-s − 1.25·23-s − 0.192·27-s + 1.48·29-s + 0.348·33-s + 1.47·37-s + 0.160·39-s − 0.937·41-s − 0.609·43-s + 0.583·47-s − 0.560·51-s + 0.662·57-s + 0.520·59-s + 1.40·61-s − 0.610·67-s + 0.722·69-s − 0.949·71-s − 0.117·73-s − 0.562·79-s + 1/9·81-s − 0.439·83-s − 0.857·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−14.66796126512561, −14.14948851867639, −13.30090918085157, −13.25195818743589, −12.38370001240135, −12.07838519185853, −11.70143475299067, −10.92398678174628, −10.52164196846684, −9.939242641682588, −9.789786309109657, −8.833010567079359, −8.212955277552267, −7.975557319219028, −7.209619480054934, −6.657781728842386, −6.100040669720364, −5.611210175736781, −5.025466694037196, −4.378048875006939, −3.936973083250112, −3.022917416192272, −2.452264294130187, −1.693780099602354, −0.8249232067907941, 0,
0.8249232067907941, 1.693780099602354, 2.452264294130187, 3.022917416192272, 3.936973083250112, 4.378048875006939, 5.025466694037196, 5.611210175736781, 6.100040669720364, 6.657781728842386, 7.209619480054934, 7.975557319219028, 8.212955277552267, 8.833010567079359, 9.789786309109657, 9.939242641682588, 10.52164196846684, 10.92398678174628, 11.70143475299067, 12.07838519185853, 12.38370001240135, 13.25195818743589, 13.30090918085157, 14.14948851867639, 14.66796126512561
