| L(s) = 1 | − 3·2-s + 2·3-s + 4-s − 6·6-s + 21·8-s − 23·9-s − 45·11-s + 2·12-s − 59·13-s − 71·16-s + 54·17-s + 69·18-s − 121·19-s + 135·22-s − 69·23-s + 42·24-s + 177·26-s − 100·27-s − 162·29-s − 88·31-s + 45·32-s − 90·33-s − 162·34-s − 23·36-s + 259·37-s + 363·38-s − 118·39-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 0.384·3-s + 1/8·4-s − 0.408·6-s + 0.928·8-s − 0.851·9-s − 1.23·11-s + 0.0481·12-s − 1.25·13-s − 1.10·16-s + 0.770·17-s + 0.903·18-s − 1.46·19-s + 1.30·22-s − 0.625·23-s + 0.357·24-s + 1.33·26-s − 0.712·27-s − 1.03·29-s − 0.509·31-s + 0.248·32-s − 0.474·33-s − 0.817·34-s − 0.106·36-s + 1.15·37-s + 1.54·38-s − 0.484·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3447568041\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3447568041\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 3 T + p^{3} T^{2} \) |
| 3 | \( 1 - 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + 45 T + p^{3} T^{2} \) |
| 13 | \( 1 + 59 T + p^{3} T^{2} \) |
| 17 | \( 1 - 54 T + p^{3} T^{2} \) |
| 19 | \( 1 + 121 T + p^{3} T^{2} \) |
| 23 | \( 1 + 3 p T + p^{3} T^{2} \) |
| 29 | \( 1 + 162 T + p^{3} T^{2} \) |
| 31 | \( 1 + 88 T + p^{3} T^{2} \) |
| 37 | \( 1 - 7 p T + p^{3} T^{2} \) |
| 41 | \( 1 - 195 T + p^{3} T^{2} \) |
| 43 | \( 1 - 286 T + p^{3} T^{2} \) |
| 47 | \( 1 + 45 T + p^{3} T^{2} \) |
| 53 | \( 1 + 597 T + p^{3} T^{2} \) |
| 59 | \( 1 + 360 T + p^{3} T^{2} \) |
| 61 | \( 1 - 392 T + p^{3} T^{2} \) |
| 67 | \( 1 - 280 T + p^{3} T^{2} \) |
| 71 | \( 1 - 48 T + p^{3} T^{2} \) |
| 73 | \( 1 + 668 T + p^{3} T^{2} \) |
| 79 | \( 1 - 782 T + p^{3} T^{2} \) |
| 83 | \( 1 + 768 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1194 T + p^{3} T^{2} \) |
| 97 | \( 1 + 902 T + p^{3} T^{2} \) |
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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
−9.376365677049136337030528622700, −8.507734516966498819698463759291, −7.82638894032474638578069066495, −7.43611706745052403283995209497, −6.03153630643110133771569337587, −5.13785186455935956075930780139, −4.15001330393488703196686324770, −2.76318250472831319611100182638, −1.99167026243351752391888723186, −0.32633284316797679934872944314,
0.32633284316797679934872944314, 1.99167026243351752391888723186, 2.76318250472831319611100182638, 4.15001330393488703196686324770, 5.13785186455935956075930780139, 6.03153630643110133771569337587, 7.43611706745052403283995209497, 7.82638894032474638578069066495, 8.507734516966498819698463759291, 9.376365677049136337030528622700
