| L(s) = 1 | + 4·5-s − 7-s − 3·9-s + 11-s − 2·13-s − 4·17-s + 6·19-s + 4·23-s + 11·25-s + 2·29-s − 2·31-s − 4·35-s − 10·37-s + 4·41-s + 8·43-s − 12·45-s + 2·47-s + 49-s − 6·53-s + 4·55-s + 12·59-s + 14·61-s + 3·63-s − 8·65-s + 12·67-s − 8·71-s + 4·73-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 0.377·7-s − 9-s + 0.301·11-s − 0.554·13-s − 0.970·17-s + 1.37·19-s + 0.834·23-s + 11/5·25-s + 0.371·29-s − 0.359·31-s − 0.676·35-s − 1.64·37-s + 0.624·41-s + 1.21·43-s − 1.78·45-s + 0.291·47-s + 1/7·49-s − 0.824·53-s + 0.539·55-s + 1.56·59-s + 1.79·61-s + 0.377·63-s − 0.992·65-s + 1.46·67-s − 0.949·71-s + 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.504821876\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.504821876\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.596108761021651344869700453421, −7.29999456543540483713986491118, −6.75534837172482153142827818885, −6.00214639283948454302408706475, −5.40821631418111626955625209230, −4.89204912487217343955499873034, −3.53751482899012863138118644990, −2.66919692702130451505800501099, −2.09312337546441837636968536211, −0.868869516443346125712398445012,
0.868869516443346125712398445012, 2.09312337546441837636968536211, 2.66919692702130451505800501099, 3.53751482899012863138118644990, 4.89204912487217343955499873034, 5.40821631418111626955625209230, 6.00214639283948454302408706475, 6.75534837172482153142827818885, 7.29999456543540483713986491118, 8.596108761021651344869700453421
