L(s) = 1 | + 3·3-s − 10.5·5-s + 9·9-s − 34.7·11-s + 37.2·13-s − 31.6·15-s + 10.5·17-s − 58.5·19-s + 125.·23-s − 13.7·25-s + 27·27-s − 35.4·29-s + 291.·31-s − 104.·33-s − 259.·37-s + 111.·39-s + 338.·41-s − 6.80·43-s − 94.9·45-s + 250.·47-s + 31.6·51-s − 536.·53-s + 366.·55-s − 175.·57-s − 35.8·59-s − 57.7·61-s − 393.·65-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.943·5-s + 0.333·9-s − 0.952·11-s + 0.795·13-s − 0.544·15-s + 0.150·17-s − 0.707·19-s + 1.13·23-s − 0.109·25-s + 0.192·27-s − 0.226·29-s + 1.69·31-s − 0.549·33-s − 1.15·37-s + 0.459·39-s + 1.28·41-s − 0.0241·43-s − 0.314·45-s + 0.778·47-s + 0.0868·51-s − 1.39·53-s + 0.898·55-s − 0.408·57-s − 0.0791·59-s − 0.121·61-s − 0.750·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 10.5T + 125T^{2} \) |
| 11 | \( 1 + 34.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 37.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 10.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 58.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 125.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 35.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 291.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 259.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 338.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 6.80T + 7.95e4T^{2} \) |
| 47 | \( 1 - 250.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 536.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 35.8T + 2.05e5T^{2} \) |
| 61 | \( 1 + 57.7T + 2.26e5T^{2} \) |
| 67 | \( 1 + 481.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 363.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 581.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 693.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.33e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 353.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.44e3T + 9.12e5T^{2} \) |
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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
−8.183708164598938593441898074661, −7.69333961758841370526986745340, −6.86178402921126138828018829689, −5.94821738587100335265558533688, −4.88422898062411965712946029257, −4.13703419761613135581947811739, −3.28202969547634036390091411165, −2.50627674639791056357748268712, −1.19484605574209766719975417736, 0,
1.19484605574209766719975417736, 2.50627674639791056357748268712, 3.28202969547634036390091411165, 4.13703419761613135581947811739, 4.88422898062411965712946029257, 5.94821738587100335265558533688, 6.86178402921126138828018829689, 7.69333961758841370526986745340, 8.183708164598938593441898074661