| L(s) = 1 | + 512·2-s − 1.38e4·3-s + 2.62e5·4-s + 4.16e6·5-s − 7.07e6·6-s + 1.33e8·7-s + 1.34e8·8-s − 9.71e8·9-s + 2.13e9·10-s + 3.79e9·11-s − 3.62e9·12-s − 4.88e9·13-s + 6.85e10·14-s − 5.75e10·15-s + 6.87e10·16-s + 3.69e11·17-s − 4.97e11·18-s + 3.95e11·19-s + 1.09e12·20-s − 1.84e12·21-s + 1.94e12·22-s + 1.21e13·23-s − 1.85e12·24-s − 1.72e12·25-s − 2.50e12·26-s + 2.94e13·27-s + 3.51e13·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.405·3-s + 0.5·4-s + 0.953·5-s − 0.286·6-s + 1.25·7-s + 0.353·8-s − 0.835·9-s + 0.674·10-s + 0.485·11-s − 0.202·12-s − 0.127·13-s + 0.886·14-s − 0.386·15-s + 0.250·16-s + 0.756·17-s − 0.591·18-s + 0.281·19-s + 0.476·20-s − 0.508·21-s + 0.343·22-s + 1.41·23-s − 0.143·24-s − 0.0905·25-s − 0.0903·26-s + 0.743·27-s + 0.627·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(\approx\) |
\(4.837515848\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.837515848\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 512T \) |
| 29 | \( 1 - 1.45e13T \) |
| good | 3 | \( 1 + 1.38e4T + 1.16e9T^{2} \) |
| 5 | \( 1 - 4.16e6T + 1.90e13T^{2} \) |
| 7 | \( 1 - 1.33e8T + 1.13e16T^{2} \) |
| 11 | \( 1 - 3.79e9T + 6.11e19T^{2} \) |
| 13 | \( 1 + 4.88e9T + 1.46e21T^{2} \) |
| 17 | \( 1 - 3.69e11T + 2.39e23T^{2} \) |
| 19 | \( 1 - 3.95e11T + 1.97e24T^{2} \) |
| 23 | \( 1 - 1.21e13T + 7.46e25T^{2} \) |
| 31 | \( 1 - 6.94e13T + 2.16e28T^{2} \) |
| 37 | \( 1 + 5.65e14T + 6.24e29T^{2} \) |
| 41 | \( 1 + 1.00e15T + 4.39e30T^{2} \) |
| 43 | \( 1 - 8.08e14T + 1.08e31T^{2} \) |
| 47 | \( 1 - 4.50e15T + 5.88e31T^{2} \) |
| 53 | \( 1 + 2.23e16T + 5.77e32T^{2} \) |
| 59 | \( 1 + 6.84e16T + 4.42e33T^{2} \) |
| 61 | \( 1 + 1.52e17T + 8.34e33T^{2} \) |
| 67 | \( 1 - 4.03e16T + 4.95e34T^{2} \) |
| 71 | \( 1 - 5.60e17T + 1.49e35T^{2} \) |
| 73 | \( 1 - 8.89e17T + 2.53e35T^{2} \) |
| 79 | \( 1 - 1.72e18T + 1.13e36T^{2} \) |
| 83 | \( 1 + 1.38e18T + 2.90e36T^{2} \) |
| 89 | \( 1 + 1.69e18T + 1.09e37T^{2} \) |
| 97 | \( 1 - 9.83e18T + 5.60e37T^{2} \) |
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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
−11.47597383384879461712681835548, −10.61866238724436691702462407255, −9.204540413918538451004074224230, −7.900616843205923185683019992274, −6.49167993492625269927096058963, −5.46519485676642670809136675318, −4.81111103846087489410688104302, −3.18648863865699150745995516034, −1.95694007764431123064009728220, −0.977833922988226175231442747176,
0.977833922988226175231442747176, 1.95694007764431123064009728220, 3.18648863865699150745995516034, 4.81111103846087489410688104302, 5.46519485676642670809136675318, 6.49167993492625269927096058963, 7.900616843205923185683019992274, 9.204540413918538451004074224230, 10.61866238724436691702462407255, 11.47597383384879461712681835548
