| L(s) = 1 | − 3-s − 5-s + 11-s + 15-s − 23-s + 27-s − 31-s − 33-s − 37-s + 2·47-s + 49-s + 2·53-s − 55-s − 59-s − 67-s + 69-s − 71-s − 81-s − 89-s + 93-s − 97-s + 2·103-s + 111-s − 113-s + 115-s + ⋯ |
| L(s) = 1 | − 3-s − 5-s + 11-s + 15-s − 23-s + 27-s − 31-s − 33-s − 37-s + 2·47-s + 49-s + 2·53-s − 55-s − 59-s − 67-s + 69-s − 71-s − 81-s − 89-s + 93-s − 97-s + 2·103-s + 111-s − 113-s + 115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3157625309\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3157625309\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + T + T^{2} \) |
| 5 | \( 1 + T + T^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )^{2} \) |
| 53 | \( ( 1 - T )^{2} \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−16.35967410986435713447548409957, −15.27319062813857186773037865264, −13.99851593145856548211536643059, −12.22485563619225556720266613180, −11.70308803032614155922865956844, −10.55715681078177439271694820025, −8.811336960685553603966194710597, −7.24907980722071614777800764527, −5.79558660706818170317355578628, −4.04199480857272067821281564908,
4.04199480857272067821281564908, 5.79558660706818170317355578628, 7.24907980722071614777800764527, 8.811336960685553603966194710597, 10.55715681078177439271694820025, 11.70308803032614155922865956844, 12.22485563619225556720266613180, 13.99851593145856548211536643059, 15.27319062813857186773037865264, 16.35967410986435713447548409957
