| L(s) = 1 | + 2-s + 4-s + 5-s + 8-s + 10-s − 2.24·11-s − 5.65·13-s + 16-s + 2.58·17-s + 6.82·19-s + 20-s − 2.24·22-s − 3.17·23-s + 25-s − 5.65·26-s − 2.58·29-s + 10.2·31-s + 32-s + 2.58·34-s − 0.242·37-s + 6.82·38-s + 40-s + 10.4·41-s + 9.07·43-s − 2.24·44-s − 3.17·46-s − 2.24·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s − 0.676·11-s − 1.56·13-s + 0.250·16-s + 0.627·17-s + 1.56·19-s + 0.223·20-s − 0.478·22-s − 0.661·23-s + 0.200·25-s − 1.10·26-s − 0.480·29-s + 1.83·31-s + 0.176·32-s + 0.443·34-s − 0.0398·37-s + 1.10·38-s + 0.158·40-s + 1.63·41-s + 1.38·43-s − 0.338·44-s − 0.467·46-s − 0.327·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.228764046\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.228764046\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2.24T + 11T^{2} \) |
| 13 | \( 1 + 5.65T + 13T^{2} \) |
| 17 | \( 1 - 2.58T + 17T^{2} \) |
| 19 | \( 1 - 6.82T + 19T^{2} \) |
| 23 | \( 1 + 3.17T + 23T^{2} \) |
| 29 | \( 1 + 2.58T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 + 0.242T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 - 9.07T + 43T^{2} \) |
| 47 | \( 1 + 2.24T + 47T^{2} \) |
| 53 | \( 1 + 0.343T + 53T^{2} \) |
| 59 | \( 1 + 3.17T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 - 9.07T + 67T^{2} \) |
| 71 | \( 1 - 4.48T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 5.17T + 83T^{2} \) |
| 89 | \( 1 + 0.828T + 89T^{2} \) |
| 97 | \( 1 - 6.48T + 97T^{2} \) |
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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
−7.970041577216843989306690371804, −7.67650417120577189564121916188, −6.87357322509909134828516681925, −5.97648325036665893431123155007, −5.31172770008620258896694634626, −4.81916269277032597488056799899, −3.82840165900630956534830222101, −2.78114868448936624469133823256, −2.32154391306376787407835723753, −0.912033113218131558545210907976,
0.912033113218131558545210907976, 2.32154391306376787407835723753, 2.78114868448936624469133823256, 3.82840165900630956534830222101, 4.81916269277032597488056799899, 5.31172770008620258896694634626, 5.97648325036665893431123155007, 6.87357322509909134828516681925, 7.67650417120577189564121916188, 7.970041577216843989306690371804
