| L(s) = 1 | − 3-s + 3.56·5-s + 7-s + 9-s + 1.56·11-s + 13-s − 3.56·15-s − 6.68·17-s + 4.68·19-s − 21-s + 5.56·23-s + 7.68·25-s − 27-s + 6.68·29-s − 6.24·31-s − 1.56·33-s + 3.56·35-s − 7.56·37-s − 39-s − 1.12·41-s + 6.43·43-s + 3.56·45-s + 49-s + 6.68·51-s + 12.2·53-s + 5.56·55-s − 4.68·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.59·5-s + 0.377·7-s + 0.333·9-s + 0.470·11-s + 0.277·13-s − 0.919·15-s − 1.62·17-s + 1.07·19-s − 0.218·21-s + 1.15·23-s + 1.53·25-s − 0.192·27-s + 1.24·29-s − 1.12·31-s − 0.271·33-s + 0.602·35-s − 1.24·37-s − 0.160·39-s − 0.175·41-s + 0.981·43-s + 0.530·45-s + 0.142·49-s + 0.936·51-s + 1.68·53-s + 0.749·55-s − 0.620·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.502435085\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.502435085\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 3.56T + 5T^{2} \) |
| 11 | \( 1 - 1.56T + 11T^{2} \) |
| 17 | \( 1 + 6.68T + 17T^{2} \) |
| 19 | \( 1 - 4.68T + 19T^{2} \) |
| 23 | \( 1 - 5.56T + 23T^{2} \) |
| 29 | \( 1 - 6.68T + 29T^{2} \) |
| 31 | \( 1 + 6.24T + 31T^{2} \) |
| 37 | \( 1 + 7.56T + 37T^{2} \) |
| 41 | \( 1 + 1.12T + 41T^{2} \) |
| 43 | \( 1 - 6.43T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 + 2.24T + 59T^{2} \) |
| 61 | \( 1 - 6.68T + 61T^{2} \) |
| 67 | \( 1 - 7.12T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 3.56T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 + 8.87T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 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
−8.779746668083853074332014555935, −7.37329235605780715955215043078, −6.75891686494518829659490246416, −6.19610596826809002492772078267, −5.34565979994984008249556320209, −4.96085807576823703487988251248, −3.90591991910765265154289396703, −2.69521575414529821230869526727, −1.84991615989452510170864099050, −0.977761983356336960621571142402,
0.977761983356336960621571142402, 1.84991615989452510170864099050, 2.69521575414529821230869526727, 3.90591991910765265154289396703, 4.96085807576823703487988251248, 5.34565979994984008249556320209, 6.19610596826809002492772078267, 6.75891686494518829659490246416, 7.37329235605780715955215043078, 8.779746668083853074332014555935
