| L(s) = 1 | + 2-s − 1.56·3-s + 4-s + 2·5-s − 1.56·6-s − 3.56·7-s + 8-s − 0.561·9-s + 2·10-s − 1.56·12-s + 3.56·13-s − 3.56·14-s − 3.12·15-s + 16-s − 3.56·17-s − 0.561·18-s + 19-s + 2·20-s + 5.56·21-s + 5.56·23-s − 1.56·24-s − 25-s + 3.56·26-s + 5.56·27-s − 3.56·28-s − 6.68·29-s − 3.12·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.901·3-s + 0.5·4-s + 0.894·5-s − 0.637·6-s − 1.34·7-s + 0.353·8-s − 0.187·9-s + 0.632·10-s − 0.450·12-s + 0.987·13-s − 0.951·14-s − 0.806·15-s + 0.250·16-s − 0.863·17-s − 0.132·18-s + 0.229·19-s + 0.447·20-s + 1.21·21-s + 1.15·23-s − 0.318·24-s − 0.200·25-s + 0.698·26-s + 1.07·27-s − 0.673·28-s − 1.24·29-s − 0.570·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.028346812\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.028346812\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 1.56T + 3T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 + 3.56T + 7T^{2} \) |
| 13 | \( 1 - 3.56T + 13T^{2} \) |
| 17 | \( 1 + 3.56T + 17T^{2} \) |
| 23 | \( 1 - 5.56T + 23T^{2} \) |
| 29 | \( 1 + 6.68T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 3.12T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + 7.80T + 53T^{2} \) |
| 59 | \( 1 - 4.68T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 - 4.68T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 2.68T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 2.24T + 83T^{2} \) |
| 89 | \( 1 + 9.12T + 89T^{2} \) |
| 97 | \( 1 - 1.12T + 97T^{2} \) |
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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
−8.388013372165264433451553011106, −7.11753857845781992089867549780, −6.60792944412009385019188220043, −5.96360662406965274345005296603, −5.65306893796962804722356948928, −4.77555400944879620371328021404, −3.75247975615199999890362818538, −3.01544449142983945833167266384, −2.07223502052819286353448144161, −0.72945090124975956079557439977,
0.72945090124975956079557439977, 2.07223502052819286353448144161, 3.01544449142983945833167266384, 3.75247975615199999890362818538, 4.77555400944879620371328021404, 5.65306893796962804722356948928, 5.96360662406965274345005296603, 6.60792944412009385019188220043, 7.11753857845781992089867549780, 8.388013372165264433451553011106
