L(s) = 1 | + 5-s + 2·11-s − 13-s + 6·17-s + 6·23-s + 25-s − 8·31-s − 6·37-s − 2·41-s − 4·43-s − 8·53-s + 2·55-s − 8·59-s + 10·61-s − 65-s + 4·67-s − 6·71-s + 14·73-s − 8·79-s + 4·83-s + 6·85-s − 10·89-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.603·11-s − 0.277·13-s + 1.45·17-s + 1.25·23-s + 1/5·25-s − 1.43·31-s − 0.986·37-s − 0.312·41-s − 0.609·43-s − 1.09·53-s + 0.269·55-s − 1.04·59-s + 1.28·61-s − 0.124·65-s + 0.488·67-s − 0.712·71-s + 1.63·73-s − 0.900·79-s + 0.439·83-s + 0.650·85-s − 1.05·89-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.06245957334554, −12.68514699905240, −12.32996267317919, −11.77999621366339, −11.30916767114170, −10.85735460263973, −10.30054304573571, −9.940801635513192, −9.294940807793181, −9.161730373203604, −8.492188219410427, −7.959816624005174, −7.468163709633942, −6.893554718913237, −6.624310223550041, −5.903391420694160, −5.350839942212068, −5.142450548086483, −4.453485607265319, −3.692301350790833, −3.344132848363989, −2.814132275259215, −1.987579848871117, −1.492358748458101, −0.9249928654932329, 0,
0.9249928654932329, 1.492358748458101, 1.987579848871117, 2.814132275259215, 3.344132848363989, 3.692301350790833, 4.453485607265319, 5.142450548086483, 5.350839942212068, 5.903391420694160, 6.624310223550041, 6.893554718913237, 7.468163709633942, 7.959816624005174, 8.492188219410427, 9.161730373203604, 9.294940807793181, 9.940801635513192, 10.30054304573571, 10.85735460263973, 11.30916767114170, 11.77999621366339, 12.32996267317919, 12.68514699905240, 13.06245957334554