L(s) = 1 | − 2·3-s + 5-s − 7-s + 9-s + 2·11-s − 2·15-s − 3·17-s − 8·19-s + 2·21-s + 6·23-s − 4·25-s + 4·27-s + 9·29-s + 6·31-s − 4·33-s − 35-s + 3·37-s + 3·41-s + 2·43-s + 45-s + 12·47-s + 49-s + 6·51-s + 5·53-s + 2·55-s + 16·57-s − 14·59-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.603·11-s − 0.516·15-s − 0.727·17-s − 1.83·19-s + 0.436·21-s + 1.25·23-s − 4/5·25-s + 0.769·27-s + 1.67·29-s + 1.07·31-s − 0.696·33-s − 0.169·35-s + 0.493·37-s + 0.468·41-s + 0.304·43-s + 0.149·45-s + 1.75·47-s + 1/7·49-s + 0.840·51-s + 0.686·53-s + 0.269·55-s + 2.11·57-s − 1.82·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.22977917472583, −13.78015887738656, −13.17889016258021, −12.79472685503285, −12.22154814856270, −11.84807287165005, −11.29399838066097, −10.79452488291532, −10.37934692350723, −9.996392374555524, −9.184214589319176, −8.768492852060837, −8.402625434851217, −7.467545265032277, −6.869804057422858, −6.398692257273034, −6.108212153528336, −5.630520982784693, −4.786138835601899, −4.416568229822763, −3.923275851797419, −2.706535946963284, −2.584434422100752, −1.470663409522489, −0.8152894624559505, 0,
0.8152894624559505, 1.470663409522489, 2.584434422100752, 2.706535946963284, 3.923275851797419, 4.416568229822763, 4.786138835601899, 5.630520982784693, 6.108212153528336, 6.398692257273034, 6.869804057422858, 7.467545265032277, 8.402625434851217, 8.768492852060837, 9.184214589319176, 9.996392374555524, 10.37934692350723, 10.79452488291532, 11.29399838066097, 11.84807287165005, 12.22154814856270, 12.79472685503285, 13.17889016258021, 13.78015887738656, 14.22977917472583