| L(s) = 1 | − 3·3-s + 22·7-s + 9·9-s + 14·11-s + 30·13-s − 62·17-s + 120·19-s − 66·21-s + 188·23-s − 27·27-s + 96·29-s − 184·31-s − 42·33-s − 406·37-s − 90·39-s + 130·41-s + 148·43-s + 448·47-s + 141·49-s + 186·51-s + 414·53-s − 360·57-s − 266·59-s − 838·61-s + 198·63-s + 248·67-s − 564·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.18·7-s + 1/3·9-s + 0.383·11-s + 0.640·13-s − 0.884·17-s + 1.44·19-s − 0.685·21-s + 1.70·23-s − 0.192·27-s + 0.614·29-s − 1.06·31-s − 0.221·33-s − 1.80·37-s − 0.369·39-s + 0.495·41-s + 0.524·43-s + 1.39·47-s + 0.411·49-s + 0.510·51-s + 1.07·53-s − 0.836·57-s − 0.586·59-s − 1.75·61-s + 0.395·63-s + 0.452·67-s − 0.984·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.295750764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.295750764\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 22 T + p^{3} T^{2} \) |
| 11 | \( 1 - 14 T + p^{3} T^{2} \) |
| 13 | \( 1 - 30 T + p^{3} T^{2} \) |
| 17 | \( 1 + 62 T + p^{3} T^{2} \) |
| 19 | \( 1 - 120 T + p^{3} T^{2} \) |
| 23 | \( 1 - 188 T + p^{3} T^{2} \) |
| 29 | \( 1 - 96 T + p^{3} T^{2} \) |
| 31 | \( 1 + 184 T + p^{3} T^{2} \) |
| 37 | \( 1 + 406 T + p^{3} T^{2} \) |
| 41 | \( 1 - 130 T + p^{3} T^{2} \) |
| 43 | \( 1 - 148 T + p^{3} T^{2} \) |
| 47 | \( 1 - 448 T + p^{3} T^{2} \) |
| 53 | \( 1 - 414 T + p^{3} T^{2} \) |
| 59 | \( 1 + 266 T + p^{3} T^{2} \) |
| 61 | \( 1 + 838 T + p^{3} T^{2} \) |
| 67 | \( 1 - 248 T + p^{3} T^{2} \) |
| 71 | \( 1 + 1020 T + p^{3} T^{2} \) |
| 73 | \( 1 + 484 T + p^{3} T^{2} \) |
| 79 | \( 1 - 48 T + p^{3} T^{2} \) |
| 83 | \( 1 - 548 T + p^{3} T^{2} \) |
| 89 | \( 1 + 650 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1816 T + p^{3} T^{2} \) |
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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
−9.122561543132764792243419390444, −8.775369801932956539882631474138, −7.53605401148281322552552766636, −7.02275973246872958295229253960, −5.88436545809590039001586462656, −5.11577234039474410407813510943, −4.37774738964240146472760864956, −3.20318630457177812341454164994, −1.74505436005947456052006303854, −0.862327830005845416480535678714,
0.862327830005845416480535678714, 1.74505436005947456052006303854, 3.20318630457177812341454164994, 4.37774738964240146472760864956, 5.11577234039474410407813510943, 5.88436545809590039001586462656, 7.02275973246872958295229253960, 7.53605401148281322552552766636, 8.775369801932956539882631474138, 9.122561543132764792243419390444
