| L(s) = 1 | − 4-s + 4·5-s − 9-s − 12·11-s + 16-s − 16·19-s − 4·20-s + 11·25-s + 8·29-s − 2·31-s + 36-s + 4·41-s + 12·44-s − 4·45-s + 10·49-s − 48·55-s − 12·59-s − 28·61-s − 64-s + 32·71-s + 16·76-s + 4·80-s + 81-s − 20·89-s − 64·95-s + 12·99-s − 11·100-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.78·5-s − 1/3·9-s − 3.61·11-s + 1/4·16-s − 3.67·19-s − 0.894·20-s + 11/5·25-s + 1.48·29-s − 0.359·31-s + 1/6·36-s + 0.624·41-s + 1.80·44-s − 0.596·45-s + 10/7·49-s − 6.47·55-s − 1.56·59-s − 3.58·61-s − 1/8·64-s + 3.79·71-s + 1.83·76-s + 0.447·80-s + 1/9·81-s − 2.11·89-s − 6.56·95-s + 1.20·99-s − 1.09·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9353984850\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9353984850\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 31 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.51562922229609008700824318812, −9.920692181020961282739520091759, −9.458820960308478406152581506333, −9.114882570523260308065727861260, −8.423286872269524714293322793995, −8.252289531777121079229109701580, −8.082992993984181268186689906233, −7.32187816876209936881109139987, −6.78361127894603513312779486550, −6.21364179357907683663796927527, −5.92531900103016535393686587314, −5.53786055272347795924294184933, −5.09057196258572380238194577904, −4.47727261522209926947945209115, −4.45309952440051840068136497396, −3.07971093944410303561855635872, −2.77623432088683007099835644420, −2.19071424827379465570161767971, −1.97375645400248099072466309148, −0.41938997965137985629949346235,
0.41938997965137985629949346235, 1.97375645400248099072466309148, 2.19071424827379465570161767971, 2.77623432088683007099835644420, 3.07971093944410303561855635872, 4.45309952440051840068136497396, 4.47727261522209926947945209115, 5.09057196258572380238194577904, 5.53786055272347795924294184933, 5.92531900103016535393686587314, 6.21364179357907683663796927527, 6.78361127894603513312779486550, 7.32187816876209936881109139987, 8.082992993984181268186689906233, 8.252289531777121079229109701580, 8.423286872269524714293322793995, 9.114882570523260308065727861260, 9.458820960308478406152581506333, 9.920692181020961282739520091759, 10.51562922229609008700824318812
