| L(s) = 1 | − 2.37·2-s − 4.44·3-s − 2.37·4-s + 19.4·5-s + 10.5·6-s + 14.9·7-s + 24.6·8-s − 7.23·9-s − 46.1·10-s + 31.1·11-s + 10.5·12-s − 5.21·13-s − 35.5·14-s − 86.4·15-s − 39.3·16-s + 17.1·18-s + 28·19-s − 46.1·20-s − 66.6·21-s − 73.8·22-s − 167.·23-s − 109.·24-s + 252.·25-s + 12.3·26-s + 152.·27-s − 35.5·28-s − 136.·29-s + ⋯ |
| L(s) = 1 | − 0.838·2-s − 0.855·3-s − 0.296·4-s + 1.73·5-s + 0.717·6-s + 0.809·7-s + 1.08·8-s − 0.267·9-s − 1.45·10-s + 0.853·11-s + 0.253·12-s − 0.111·13-s − 0.678·14-s − 1.48·15-s − 0.615·16-s + 0.224·18-s + 0.338·19-s − 0.515·20-s − 0.692·21-s − 0.715·22-s − 1.51·23-s − 0.930·24-s + 2.02·25-s + 0.0932·26-s + 1.08·27-s − 0.240·28-s − 0.871·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.155244576\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.155244576\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + 2.37T + 8T^{2} \) |
| 3 | \( 1 + 4.44T + 27T^{2} \) |
| 5 | \( 1 - 19.4T + 125T^{2} \) |
| 7 | \( 1 - 14.9T + 343T^{2} \) |
| 11 | \( 1 - 31.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 5.21T + 2.19e3T^{2} \) |
| 19 | \( 1 - 28T + 6.85e3T^{2} \) |
| 23 | \( 1 + 167.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 136.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 50.5T + 2.97e4T^{2} \) |
| 37 | \( 1 - 260.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 183.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 348.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 318.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 408.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 108.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 123.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 243.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 42.7T + 3.57e5T^{2} \) |
| 73 | \( 1 + 875.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 750.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 472.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 376.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 303.T + 9.12e5T^{2} \) |
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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
−11.09793839032658422408929641053, −10.31037649484855992917266898426, −9.491273298835868927818445720496, −8.813270728296891197903276088460, −7.60219298169262956138692606002, −6.19461160953782282007185424526, −5.55334560943517907656552381421, −4.43697187486327925763906908139, −2.08345310794840589371117501554, −0.952937937092653269311823973128,
0.952937937092653269311823973128, 2.08345310794840589371117501554, 4.43697187486327925763906908139, 5.55334560943517907656552381421, 6.19461160953782282007185424526, 7.60219298169262956138692606002, 8.813270728296891197903276088460, 9.491273298835868927818445720496, 10.31037649484855992917266898426, 11.09793839032658422408929641053
