| L(s) = 1 | + 2-s + 4-s + 2·5-s − 2·7-s + 8-s + 2·10-s + 4·11-s − 13-s − 2·14-s + 16-s − 6·19-s + 2·20-s + 4·22-s − 4·23-s − 25-s − 26-s − 2·28-s + 8·29-s − 2·31-s + 32-s − 4·35-s + 6·37-s − 6·38-s + 2·40-s − 6·41-s − 8·43-s + 4·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.755·7-s + 0.353·8-s + 0.632·10-s + 1.20·11-s − 0.277·13-s − 0.534·14-s + 1/4·16-s − 1.37·19-s + 0.447·20-s + 0.852·22-s − 0.834·23-s − 1/5·25-s − 0.196·26-s − 0.377·28-s + 1.48·29-s − 0.359·31-s + 0.176·32-s − 0.676·35-s + 0.986·37-s − 0.973·38-s + 0.316·40-s − 0.937·41-s − 1.21·43-s + 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.955366556\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.955366556\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−12.37050893630544855392229401962, −11.39787343056409573124340982210, −10.15704375173210575584075343151, −9.511260567162705652960388658514, −8.241527102852606326983085030222, −6.49638165287752346862203973768, −6.33856513547641234727266817168, −4.80645914681781210957697738257, −3.53791924823882088462258917674, −2.01663812390586556648231292076,
2.01663812390586556648231292076, 3.53791924823882088462258917674, 4.80645914681781210957697738257, 6.33856513547641234727266817168, 6.49638165287752346862203973768, 8.241527102852606326983085030222, 9.511260567162705652960388658514, 10.15704375173210575584075343151, 11.39787343056409573124340982210, 12.37050893630544855392229401962
