| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s + 2.44·11-s − 3.44·13-s − 14-s + 16-s + 17-s + 0.449·19-s + 2.44·22-s − 3.44·23-s − 3.44·26-s − 28-s − 7.44·29-s + 3.44·31-s + 32-s + 34-s − 11.3·37-s + 0.449·38-s + 4.89·41-s + 5.89·43-s + 2.44·44-s − 3.44·46-s − 7.79·47-s + 49-s − 3.44·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.377·7-s + 0.353·8-s + 0.738·11-s − 0.956·13-s − 0.267·14-s + 0.250·16-s + 0.242·17-s + 0.103·19-s + 0.522·22-s − 0.719·23-s − 0.676·26-s − 0.188·28-s − 1.38·29-s + 0.619·31-s + 0.176·32-s + 0.171·34-s − 1.86·37-s + 0.0729·38-s + 0.765·41-s + 0.899·43-s + 0.369·44-s − 0.508·46-s − 1.13·47-s + 0.142·49-s − 0.478·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 - 2.44T + 11T^{2} \) |
| 13 | \( 1 + 3.44T + 13T^{2} \) |
| 17 | \( 1 - T + 17T^{2} \) |
| 19 | \( 1 - 0.449T + 19T^{2} \) |
| 23 | \( 1 + 3.44T + 23T^{2} \) |
| 29 | \( 1 + 7.44T + 29T^{2} \) |
| 31 | \( 1 - 3.44T + 31T^{2} \) |
| 37 | \( 1 + 11.3T + 37T^{2} \) |
| 41 | \( 1 - 4.89T + 41T^{2} \) |
| 43 | \( 1 - 5.89T + 43T^{2} \) |
| 47 | \( 1 + 7.79T + 47T^{2} \) |
| 53 | \( 1 + 0.550T + 53T^{2} \) |
| 59 | \( 1 + 9.89T + 59T^{2} \) |
| 61 | \( 1 - 1.10T + 61T^{2} \) |
| 67 | \( 1 + 5T + 67T^{2} \) |
| 71 | \( 1 - 0.550T + 71T^{2} \) |
| 73 | \( 1 - 7.79T + 73T^{2} \) |
| 79 | \( 1 + 1.55T + 79T^{2} \) |
| 83 | \( 1 + 1.10T + 83T^{2} \) |
| 89 | \( 1 + 11T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−7.25235717032043167988560551240, −6.62373508989685694367545855441, −5.93690824051396671841798556188, −5.31046390515458387469956022780, −4.53280816433302398377706379506, −3.84216370159458564912897245354, −3.17362689453215552763616118929, −2.29079002649162862105070764390, −1.44163522678942335387059079644, 0,
1.44163522678942335387059079644, 2.29079002649162862105070764390, 3.17362689453215552763616118929, 3.84216370159458564912897245354, 4.53280816433302398377706379506, 5.31046390515458387469956022780, 5.93690824051396671841798556188, 6.62373508989685694367545855441, 7.25235717032043167988560551240
