| L(s) = 1 | − 4-s − 2·7-s + 16-s + 2·19-s + 25-s + 2·28-s − 2·31-s + 2·37-s + 2·43-s + 3·49-s − 64-s − 2·73-s − 2·76-s − 100-s + 2·109-s − 2·112-s + ⋯ |
| L(s) = 1 | − 4-s − 2·7-s + 16-s + 2·19-s + 25-s + 2·28-s − 2·31-s + 2·37-s + 2·43-s + 3·49-s − 64-s − 2·73-s − 2·76-s − 100-s + 2·109-s − 2·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7020481518\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7020481518\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 223 | \( 1 + T \) |
| good | 2 | \( 1 + T^{2} \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( ( 1 + T )^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( ( 1 - T )^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( ( 1 + T )^{2} \) |
| 37 | \( ( 1 - T )^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( ( 1 - T )^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 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
−9.295483737652050024558347200814, −8.978057908770733751238845670554, −7.65262350123440696539272297607, −7.11295277588960006563500878245, −5.98338243537363562300314601058, −5.51624408186088424564462063020, −4.33688388934765480097812841036, −3.46575196039347532070114231535, −2.85010580782358252151196403436, −0.829167485574491530653572894879,
0.829167485574491530653572894879, 2.85010580782358252151196403436, 3.46575196039347532070114231535, 4.33688388934765480097812841036, 5.51624408186088424564462063020, 5.98338243537363562300314601058, 7.11295277588960006563500878245, 7.65262350123440696539272297607, 8.978057908770733751238845670554, 9.295483737652050024558347200814
