| L(s) = 1 | − 3-s + 2.19·5-s + 4.05·7-s + 9-s + 5.80·11-s − 1.06·13-s − 2.19·15-s − 6.66·17-s + 5.93·19-s − 4.05·21-s − 23-s − 0.177·25-s − 27-s − 29-s − 2.78·31-s − 5.80·33-s + 8.90·35-s − 2.24·37-s + 1.06·39-s + 5.88·41-s + 9.32·43-s + 2.19·45-s + 10.5·47-s + 9.42·49-s + 6.66·51-s − 0.135·53-s + 12.7·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.982·5-s + 1.53·7-s + 0.333·9-s + 1.75·11-s − 0.296·13-s − 0.567·15-s − 1.61·17-s + 1.36·19-s − 0.884·21-s − 0.208·23-s − 0.0354·25-s − 0.192·27-s − 0.185·29-s − 0.499·31-s − 1.01·33-s + 1.50·35-s − 0.369·37-s + 0.171·39-s + 0.918·41-s + 1.42·43-s + 0.327·45-s + 1.54·47-s + 1.34·49-s + 0.932·51-s − 0.0185·53-s + 1.71·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.948492710\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.948492710\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 - 2.19T + 5T^{2} \) |
| 7 | \( 1 - 4.05T + 7T^{2} \) |
| 11 | \( 1 - 5.80T + 11T^{2} \) |
| 13 | \( 1 + 1.06T + 13T^{2} \) |
| 17 | \( 1 + 6.66T + 17T^{2} \) |
| 19 | \( 1 - 5.93T + 19T^{2} \) |
| 31 | \( 1 + 2.78T + 31T^{2} \) |
| 37 | \( 1 + 2.24T + 37T^{2} \) |
| 41 | \( 1 - 5.88T + 41T^{2} \) |
| 43 | \( 1 - 9.32T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 + 0.135T + 53T^{2} \) |
| 59 | \( 1 + 9.02T + 59T^{2} \) |
| 61 | \( 1 + 14.8T + 61T^{2} \) |
| 67 | \( 1 - 2.71T + 67T^{2} \) |
| 71 | \( 1 - 9.01T + 71T^{2} \) |
| 73 | \( 1 + 2.61T + 73T^{2} \) |
| 79 | \( 1 + 5.08T + 79T^{2} \) |
| 83 | \( 1 - 4.96T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 - 3.45T + 97T^{2} \) |
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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
−7.55119019301605067043916519398, −7.26027890844087682751019201301, −6.17344121109009194518797145596, −5.94052542679436223716275515692, −4.96350724397541312002984803292, −4.50104447945318442513372353876, −3.72469082515530207237371191888, −2.31607435579947665969014334294, −1.71378789821569887492024735958, −0.954782959873300926740108766866,
0.954782959873300926740108766866, 1.71378789821569887492024735958, 2.31607435579947665969014334294, 3.72469082515530207237371191888, 4.50104447945318442513372353876, 4.96350724397541312002984803292, 5.94052542679436223716275515692, 6.17344121109009194518797145596, 7.26027890844087682751019201301, 7.55119019301605067043916519398
