| L(s) = 1 | + 3-s + 2.60·7-s + 9-s − 5.61·11-s + 13-s + 5.77·17-s − 0.234·19-s + 2.60·21-s + 7.01·23-s + 27-s + 4.20·29-s + 4.41·31-s − 5.61·33-s − 2.96·37-s + 39-s − 10.9·41-s + 0.188·43-s + 7.84·47-s − 0.234·49-s + 5.77·51-s + 11.9·53-s − 0.234·57-s − 12.1·59-s − 2.98·61-s + 2.60·63-s + 3.56·67-s + 7.01·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.983·7-s + 0.333·9-s − 1.69·11-s + 0.277·13-s + 1.40·17-s − 0.0537·19-s + 0.567·21-s + 1.46·23-s + 0.192·27-s + 0.780·29-s + 0.792·31-s − 0.977·33-s − 0.487·37-s + 0.160·39-s − 1.71·41-s + 0.0287·43-s + 1.14·47-s − 0.0334·49-s + 0.809·51-s + 1.64·53-s − 0.0310·57-s − 1.58·59-s − 0.381·61-s + 0.327·63-s + 0.436·67-s + 0.844·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.966171062\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.966171062\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 2.60T + 7T^{2} \) |
| 11 | \( 1 + 5.61T + 11T^{2} \) |
| 17 | \( 1 - 5.77T + 17T^{2} \) |
| 19 | \( 1 + 0.234T + 19T^{2} \) |
| 23 | \( 1 - 7.01T + 23T^{2} \) |
| 29 | \( 1 - 4.20T + 29T^{2} \) |
| 31 | \( 1 - 4.41T + 31T^{2} \) |
| 37 | \( 1 + 2.96T + 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 - 0.188T + 43T^{2} \) |
| 47 | \( 1 - 7.84T + 47T^{2} \) |
| 53 | \( 1 - 11.9T + 53T^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 + 2.98T + 61T^{2} \) |
| 67 | \( 1 - 3.56T + 67T^{2} \) |
| 71 | \( 1 - 3.76T + 71T^{2} \) |
| 73 | \( 1 + 14.5T + 73T^{2} \) |
| 79 | \( 1 + 4.96T + 79T^{2} \) |
| 83 | \( 1 - 3.39T + 83T^{2} \) |
| 89 | \( 1 - 17.2T + 89T^{2} \) |
| 97 | \( 1 + 5.20T + 97T^{2} \) |
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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
−7.77086855932683396350066399202, −7.50579412013665620767526680097, −6.57860587091515752273206636918, −5.50347107945682503032333749406, −5.10398295918201633320956077844, −4.41920179953936698240972233904, −3.26982534816465364611938533866, −2.81295908393322705707309386519, −1.81654394941875427426578183329, −0.862649054206748742484748454290,
0.862649054206748742484748454290, 1.81654394941875427426578183329, 2.81295908393322705707309386519, 3.26982534816465364611938533866, 4.41920179953936698240972233904, 5.10398295918201633320956077844, 5.50347107945682503032333749406, 6.57860587091515752273206636918, 7.50579412013665620767526680097, 7.77086855932683396350066399202
