L(s) = 1 | + 7.29·3-s + 8.24·5-s + 11.0·7-s + 26.2·9-s + 53.4·11-s − 0.0793·13-s + 60.1·15-s − 34.4·19-s + 80.4·21-s − 113.·23-s − 57.0·25-s − 5.52·27-s + 154.·29-s + 20.1·31-s + 389.·33-s + 90.8·35-s − 18.4·37-s − 0.578·39-s + 391.·41-s + 330.·43-s + 216.·45-s + 404.·47-s − 221.·49-s + 461.·53-s + 440.·55-s − 251.·57-s + 807.·59-s + ⋯ |
L(s) = 1 | + 1.40·3-s + 0.737·5-s + 0.594·7-s + 0.971·9-s + 1.46·11-s − 0.00169·13-s + 1.03·15-s − 0.416·19-s + 0.835·21-s − 1.02·23-s − 0.456·25-s − 0.0393·27-s + 0.986·29-s + 0.116·31-s + 2.05·33-s + 0.438·35-s − 0.0820·37-s − 0.00237·39-s + 1.49·41-s + 1.17·43-s + 0.716·45-s + 1.25·47-s − 0.645·49-s + 1.19·53-s + 1.07·55-s − 0.584·57-s + 1.78·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(5.642739884\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.642739884\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
good | 3 | \( 1 - 7.29T + 27T^{2} \) |
| 5 | \( 1 - 8.24T + 125T^{2} \) |
| 7 | \( 1 - 11.0T + 343T^{2} \) |
| 11 | \( 1 - 53.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 0.0793T + 2.19e3T^{2} \) |
| 19 | \( 1 + 34.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 113.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 154.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 20.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 18.4T + 5.06e4T^{2} \) |
| 41 | \( 1 - 391.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 330.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 404.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 461.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 807.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 413.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 85.0T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.07e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 282.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.16e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 365.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 274.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 942.T + 9.12e5T^{2} \) |
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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
−8.681782824059562539993965677125, −8.076754118706782691982830649799, −7.26338641141921927982273189457, −6.34987835831818265082735238134, −5.61317302447418623796198808029, −4.27244973287131077621961739275, −3.86427885826376491300228466816, −2.60731914127513058954565218224, −2.00154948284453380576484859892, −1.06019571479251481284949848204,
1.06019571479251481284949848204, 2.00154948284453380576484859892, 2.60731914127513058954565218224, 3.86427885826376491300228466816, 4.27244973287131077621961739275, 5.61317302447418623796198808029, 6.34987835831818265082735238134, 7.26338641141921927982273189457, 8.076754118706782691982830649799, 8.681782824059562539993965677125