L(s) = 1 | + 1.83·3-s − 3.97·7-s + 0.384·9-s + 2.10·11-s + 5.35·13-s + 1.29·17-s + 2.10·19-s − 7.30·21-s − 23-s − 4.81·27-s + 6.03·29-s − 8.32·31-s + 3.86·33-s + 5.10·37-s + 9.85·39-s − 8.33·41-s + 7.78·43-s + 11.3·47-s + 8.78·49-s + 2.38·51-s − 0.573·53-s + 3.86·57-s − 9.17·59-s + 13.5·61-s − 1.52·63-s − 15.8·67-s − 1.83·69-s + ⋯ |
L(s) = 1 | + 1.06·3-s − 1.50·7-s + 0.128·9-s + 0.634·11-s + 1.48·13-s + 0.314·17-s + 0.482·19-s − 1.59·21-s − 0.208·23-s − 0.926·27-s + 1.11·29-s − 1.49·31-s + 0.673·33-s + 0.838·37-s + 1.57·39-s − 1.30·41-s + 1.18·43-s + 1.66·47-s + 1.25·49-s + 0.333·51-s − 0.0787·53-s + 0.512·57-s − 1.19·59-s + 1.73·61-s − 0.192·63-s − 1.93·67-s − 0.221·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.551332857\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.551332857\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 - 1.83T + 3T^{2} \) |
| 7 | \( 1 + 3.97T + 7T^{2} \) |
| 11 | \( 1 - 2.10T + 11T^{2} \) |
| 13 | \( 1 - 5.35T + 13T^{2} \) |
| 17 | \( 1 - 1.29T + 17T^{2} \) |
| 19 | \( 1 - 2.10T + 19T^{2} \) |
| 29 | \( 1 - 6.03T + 29T^{2} \) |
| 31 | \( 1 + 8.32T + 31T^{2} \) |
| 37 | \( 1 - 5.10T + 37T^{2} \) |
| 41 | \( 1 + 8.33T + 41T^{2} \) |
| 43 | \( 1 - 7.78T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 + 0.573T + 53T^{2} \) |
| 59 | \( 1 + 9.17T + 59T^{2} \) |
| 61 | \( 1 - 13.5T + 61T^{2} \) |
| 67 | \( 1 + 15.8T + 67T^{2} \) |
| 71 | \( 1 - 14.9T + 71T^{2} \) |
| 73 | \( 1 - 8.36T + 73T^{2} \) |
| 79 | \( 1 + 9.41T + 79T^{2} \) |
| 83 | \( 1 - 1.51T + 83T^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 + 0.337T + 97T^{2} \) |
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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
−8.496986063375097438588073201233, −7.65774249854845684838945129821, −6.89997405445086573112836953923, −6.14470501272361008750308374417, −5.65065003750068433573606536440, −4.22688832558102923691103565392, −3.48593461368859395516351503268, −3.16981120625340703592729183770, −2.10503048406690081825881614053, −0.848774523513021746028498612602,
0.848774523513021746028498612602, 2.10503048406690081825881614053, 3.16981120625340703592729183770, 3.48593461368859395516351503268, 4.22688832558102923691103565392, 5.65065003750068433573606536440, 6.14470501272361008750308374417, 6.89997405445086573112836953923, 7.65774249854845684838945129821, 8.496986063375097438588073201233