L(s) = 1 | − 3-s + 5-s + 4.11·7-s + 9-s + 2.11·11-s + 4.80·13-s − 15-s + 4·17-s − 19-s − 4.11·21-s + 0.691·23-s + 25-s − 27-s + 4.11·29-s + 2·31-s − 2.11·33-s + 4.11·35-s + 3.42·37-s − 4.80·39-s + 5.42·41-s + 2.80·43-s + 45-s − 0.691·47-s + 9.91·49-s − 4·51-s − 2.69·53-s + 2.11·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.55·7-s + 0.333·9-s + 0.637·11-s + 1.33·13-s − 0.258·15-s + 0.970·17-s − 0.229·19-s − 0.897·21-s + 0.144·23-s + 0.200·25-s − 0.192·27-s + 0.763·29-s + 0.359·31-s − 0.367·33-s + 0.695·35-s + 0.562·37-s − 0.769·39-s + 0.846·41-s + 0.427·43-s + 0.149·45-s − 0.100·47-s + 1.41·49-s − 0.560·51-s − 0.369·53-s + 0.284·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.674518033\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.674518033\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 - 4.11T + 7T^{2} \) |
| 11 | \( 1 - 2.11T + 11T^{2} \) |
| 13 | \( 1 - 4.80T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 23 | \( 1 - 0.691T + 23T^{2} \) |
| 29 | \( 1 - 4.11T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 3.42T + 37T^{2} \) |
| 41 | \( 1 - 5.42T + 41T^{2} \) |
| 43 | \( 1 - 2.80T + 43T^{2} \) |
| 47 | \( 1 + 0.691T + 47T^{2} \) |
| 53 | \( 1 + 2.69T + 53T^{2} \) |
| 59 | \( 1 + 9.53T + 59T^{2} \) |
| 61 | \( 1 + 2.91T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 5.60T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 15.1T + 79T^{2} \) |
| 83 | \( 1 + 6.22T + 83T^{2} \) |
| 89 | \( 1 + 1.42T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−8.243593692480374962516431046751, −7.73181205847319178499869699827, −6.75751791862380277356311734273, −6.01883073793229236287095203401, −5.49561365059393983937230213453, −4.60362157323689074996289912652, −4.03662244302245061224126887879, −2.84643309068377165297398054818, −1.54321547353715918194906029431, −1.12402033041330793032536183521,
1.12402033041330793032536183521, 1.54321547353715918194906029431, 2.84643309068377165297398054818, 4.03662244302245061224126887879, 4.60362157323689074996289912652, 5.49561365059393983937230213453, 6.01883073793229236287095203401, 6.75751791862380277356311734273, 7.73181205847319178499869699827, 8.243593692480374962516431046751