L(s) = 1 | − 4-s − 5-s + 9-s − 11-s + 16-s + 20-s + 25-s − 2·31-s − 36-s + 44-s − 45-s − 49-s + 55-s + 2·59-s − 64-s + 2·71-s − 80-s + 81-s − 2·89-s − 99-s − 100-s + ⋯ |
L(s) = 1 | − 4-s − 5-s + 9-s − 11-s + 16-s + 20-s + 25-s − 2·31-s − 36-s + 44-s − 45-s − 49-s + 55-s + 2·59-s − 64-s + 2·71-s − 80-s + 81-s − 2·89-s − 99-s − 100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3736291621\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3736291621\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 + T^{2} \) |
| 3 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( 1 + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 + T )^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( ( 1 + T )^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−15.59186099364208642110354093174, −14.56717603366310796920991740421, −13.16149603660972941306714478301, −12.50918611307147520131121898334, −10.94513953936249769419384576241, −9.733624443992794679615252048198, −8.364696256624625196471110219171, −7.30588121583844319826286747084, −5.10842423248229864558432670603, −3.81577785438201065364975996334,
3.81577785438201065364975996334, 5.10842423248229864558432670603, 7.30588121583844319826286747084, 8.364696256624625196471110219171, 9.733624443992794679615252048198, 10.94513953936249769419384576241, 12.50918611307147520131121898334, 13.16149603660972941306714478301, 14.56717603366310796920991740421, 15.59186099364208642110354093174