L(s) = 1 | + 0.585·5-s + 3.69·7-s + 4.14·11-s − 3.41·13-s − 2.82·17-s + 6.30·19-s + 6.75·23-s − 4.65·25-s + 7.41·29-s + 3.06·31-s + 2.16·35-s − 9.07·37-s + 4·41-s + 1.08·43-s − 3.06·47-s + 6.65·49-s + 4.58·53-s + 2.42·55-s + 1.08·59-s + 1.07·61-s − 2·65-s − 1.97·67-s + 8.02·71-s − 6.48·73-s + 15.3·77-s − 14.7·79-s + 13.6·83-s + ⋯ |
L(s) = 1 | + 0.261·5-s + 1.39·7-s + 1.24·11-s − 0.946·13-s − 0.685·17-s + 1.44·19-s + 1.40·23-s − 0.931·25-s + 1.37·29-s + 0.549·31-s + 0.365·35-s − 1.49·37-s + 0.624·41-s + 0.165·43-s − 0.446·47-s + 0.950·49-s + 0.629·53-s + 0.327·55-s + 0.140·59-s + 0.137·61-s − 0.248·65-s − 0.241·67-s + 0.952·71-s − 0.759·73-s + 1.74·77-s − 1.66·79-s + 1.50·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.095075109\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.095075109\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 0.585T + 5T^{2} \) |
| 7 | \( 1 - 3.69T + 7T^{2} \) |
| 11 | \( 1 - 4.14T + 11T^{2} \) |
| 13 | \( 1 + 3.41T + 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 - 6.30T + 19T^{2} \) |
| 23 | \( 1 - 6.75T + 23T^{2} \) |
| 29 | \( 1 - 7.41T + 29T^{2} \) |
| 31 | \( 1 - 3.06T + 31T^{2} \) |
| 37 | \( 1 + 9.07T + 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 - 1.08T + 43T^{2} \) |
| 47 | \( 1 + 3.06T + 47T^{2} \) |
| 53 | \( 1 - 4.58T + 53T^{2} \) |
| 59 | \( 1 - 1.08T + 59T^{2} \) |
| 61 | \( 1 - 1.07T + 61T^{2} \) |
| 67 | \( 1 + 1.97T + 67T^{2} \) |
| 71 | \( 1 - 8.02T + 71T^{2} \) |
| 73 | \( 1 + 6.48T + 73T^{2} \) |
| 79 | \( 1 + 14.7T + 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 + 4.82T + 89T^{2} \) |
| 97 | \( 1 - 5.17T + 97T^{2} \) |
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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
−7.60387094126565903666454149938, −7.12222565493543744131544676085, −6.44976771600785105531422039964, −5.51245643345618831060533296474, −4.89431044042490087014278762260, −4.43421999045745900996774178152, −3.44978020427279514449331512107, −2.52602173933100596976584331262, −1.64935330006008947707760524552, −0.920265768119898167165551029765,
0.920265768119898167165551029765, 1.64935330006008947707760524552, 2.52602173933100596976584331262, 3.44978020427279514449331512107, 4.43421999045745900996774178152, 4.89431044042490087014278762260, 5.51245643345618831060533296474, 6.44976771600785105531422039964, 7.12222565493543744131544676085, 7.60387094126565903666454149938