L(s) = 1 | + 2-s + 4-s + 2.60·7-s + 8-s + 6.30·11-s + 5.30·13-s + 2.60·14-s + 16-s + 4.60·17-s + 0.697·19-s + 6.30·22-s − 4·23-s − 5·25-s + 5.30·26-s + 2.60·28-s + 1.69·29-s − 0.605·31-s + 32-s + 4.60·34-s − 2.60·37-s + 0.697·38-s − 4·41-s − 6.90·43-s + 6.30·44-s − 4·46-s + 4.30·47-s − 0.211·49-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.984·7-s + 0.353·8-s + 1.90·11-s + 1.47·13-s + 0.696·14-s + 0.250·16-s + 1.11·17-s + 0.159·19-s + 1.34·22-s − 0.834·23-s − 25-s + 1.03·26-s + 0.492·28-s + 0.315·29-s − 0.108·31-s + 0.176·32-s + 0.789·34-s − 0.428·37-s + 0.113·38-s − 0.624·41-s − 1.05·43-s + 0.950·44-s − 0.589·46-s + 0.627·47-s − 0.0301·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.338704375\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.338704375\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 223 | \( 1 + T \) |
good | 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 2.60T + 7T^{2} \) |
| 11 | \( 1 - 6.30T + 11T^{2} \) |
| 13 | \( 1 - 5.30T + 13T^{2} \) |
| 17 | \( 1 - 4.60T + 17T^{2} \) |
| 19 | \( 1 - 0.697T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 1.69T + 29T^{2} \) |
| 31 | \( 1 + 0.605T + 31T^{2} \) |
| 37 | \( 1 + 2.60T + 37T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 + 6.90T + 43T^{2} \) |
| 47 | \( 1 - 4.30T + 47T^{2} \) |
| 53 | \( 1 + 3.90T + 53T^{2} \) |
| 59 | \( 1 + 3.30T + 59T^{2} \) |
| 61 | \( 1 + 6.30T + 61T^{2} \) |
| 67 | \( 1 + 2.60T + 67T^{2} \) |
| 71 | \( 1 + 3.21T + 71T^{2} \) |
| 73 | \( 1 - 1.90T + 73T^{2} \) |
| 79 | \( 1 + 9.51T + 79T^{2} \) |
| 83 | \( 1 - 9.21T + 83T^{2} \) |
| 89 | \( 1 - 2.78T + 89T^{2} \) |
| 97 | \( 1 + 15.2T + 97T^{2} \) |
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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
−8.327948808020895528790761162709, −7.75829950159078180381952489958, −6.78108393961842253342077941767, −6.13742942172902619036742078109, −5.54958076003078191977393141470, −4.54032896007714357708221393900, −3.85513249980711232195938008799, −3.30996906844965300312688807378, −1.74274490168439221789530784475, −1.29875341973171179208727592951,
1.29875341973171179208727592951, 1.74274490168439221789530784475, 3.30996906844965300312688807378, 3.85513249980711232195938008799, 4.54032896007714357708221393900, 5.54958076003078191977393141470, 6.13742942172902619036742078109, 6.78108393961842253342077941767, 7.75829950159078180381952489958, 8.327948808020895528790761162709