L(s) = 1 | − 0.732·3-s − 5-s − 2.73·7-s − 2.46·9-s + 2·11-s − 3.46·13-s + 0.732·15-s − 3.46·17-s + 7.46·19-s + 2·21-s + 4.19·23-s + 25-s + 4·27-s + 6.92·29-s − 1.46·31-s − 1.46·33-s + 2.73·35-s − 2·37-s + 2.53·39-s + 5.46·41-s + 8.73·43-s + 2.46·45-s − 6.73·47-s + 0.464·49-s + 2.53·51-s + 4.53·53-s − 2·55-s + ⋯ |
L(s) = 1 | − 0.422·3-s − 0.447·5-s − 1.03·7-s − 0.821·9-s + 0.603·11-s − 0.960·13-s + 0.189·15-s − 0.840·17-s + 1.71·19-s + 0.436·21-s + 0.874·23-s + 0.200·25-s + 0.769·27-s + 1.28·29-s − 0.262·31-s − 0.254·33-s + 0.461·35-s − 0.328·37-s + 0.406·39-s + 0.853·41-s + 1.33·43-s + 0.367·45-s − 0.981·47-s + 0.0663·49-s + 0.355·51-s + 0.623·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9128753685\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9128753685\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
good | 3 | \( 1 + 0.732T + 3T^{2} \) |
| 7 | \( 1 + 2.73T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 - 7.46T + 19T^{2} \) |
| 23 | \( 1 - 4.19T + 23T^{2} \) |
| 29 | \( 1 - 6.92T + 29T^{2} \) |
| 31 | \( 1 + 1.46T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 5.46T + 41T^{2} \) |
| 43 | \( 1 - 8.73T + 43T^{2} \) |
| 47 | \( 1 + 6.73T + 47T^{2} \) |
| 53 | \( 1 - 4.53T + 53T^{2} \) |
| 59 | \( 1 - 0.535T + 59T^{2} \) |
| 61 | \( 1 + 4.92T + 61T^{2} \) |
| 67 | \( 1 - 7.26T + 67T^{2} \) |
| 71 | \( 1 + 1.46T + 71T^{2} \) |
| 73 | \( 1 + 0.535T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 - 4.73T + 83T^{2} \) |
| 89 | \( 1 - 4.92T + 89T^{2} \) |
| 97 | \( 1 - 6.39T + 97T^{2} \) |
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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
−9.485390641777359016768945726897, −9.106596460467922606661847011570, −7.994308452090982718640075283941, −7.04798069647412296691920975021, −6.46542553214848439283986566875, −5.46888555848031269390399544413, −4.63190343746558618125675085263, −3.41127724929545797944794431389, −2.65589433541991244572373284709, −0.69756775374337419424295897806,
0.69756775374337419424295897806, 2.65589433541991244572373284709, 3.41127724929545797944794431389, 4.63190343746558618125675085263, 5.46888555848031269390399544413, 6.46542553214848439283986566875, 7.04798069647412296691920975021, 7.994308452090982718640075283941, 9.106596460467922606661847011570, 9.485390641777359016768945726897