| L(s) = 1 | − 2.75·3-s + 0.760·5-s + 0.427·7-s + 4.60·9-s − 3.25·11-s − 6.57·13-s − 2.09·15-s + 17-s − 0.578·19-s − 1.17·21-s − 2.85·23-s − 4.42·25-s − 4.43·27-s − 6.79·29-s + 9.48·31-s + 8.97·33-s + 0.324·35-s + 6.42·37-s + 18.1·39-s − 12.1·41-s + 0.645·43-s + 3.50·45-s + 12.2·47-s − 6.81·49-s − 2.75·51-s + 6.10·53-s − 2.47·55-s + ⋯ |
| L(s) = 1 | − 1.59·3-s + 0.339·5-s + 0.161·7-s + 1.53·9-s − 0.980·11-s − 1.82·13-s − 0.541·15-s + 0.242·17-s − 0.132·19-s − 0.257·21-s − 0.594·23-s − 0.884·25-s − 0.853·27-s − 1.26·29-s + 1.70·31-s + 1.56·33-s + 0.0548·35-s + 1.05·37-s + 2.90·39-s − 1.89·41-s + 0.0984·43-s + 0.522·45-s + 1.78·47-s − 0.973·49-s − 0.386·51-s + 0.838·53-s − 0.333·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5381115051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5381115051\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 + 2.75T + 3T^{2} \) |
| 5 | \( 1 - 0.760T + 5T^{2} \) |
| 7 | \( 1 - 0.427T + 7T^{2} \) |
| 11 | \( 1 + 3.25T + 11T^{2} \) |
| 13 | \( 1 + 6.57T + 13T^{2} \) |
| 19 | \( 1 + 0.578T + 19T^{2} \) |
| 23 | \( 1 + 2.85T + 23T^{2} \) |
| 29 | \( 1 + 6.79T + 29T^{2} \) |
| 31 | \( 1 - 9.48T + 31T^{2} \) |
| 37 | \( 1 - 6.42T + 37T^{2} \) |
| 41 | \( 1 + 12.1T + 41T^{2} \) |
| 43 | \( 1 - 0.645T + 43T^{2} \) |
| 47 | \( 1 - 12.2T + 47T^{2} \) |
| 53 | \( 1 - 6.10T + 53T^{2} \) |
| 61 | \( 1 + 4.26T + 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 - 16.0T + 71T^{2} \) |
| 73 | \( 1 + 6.67T + 73T^{2} \) |
| 79 | \( 1 - 0.0868T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 + 4.56T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−8.241502331889372565439388543213, −7.54245964266327663708826214166, −6.93681868022287711421127559494, −6.01221882342850302089762017150, −5.51618626742327860563072320775, −4.88609856105951209797534173292, −4.23698579650760407945864370138, −2.78198254310737045628424409153, −1.87399539089503036047489264611, −0.43800267192033771837738730116,
0.43800267192033771837738730116, 1.87399539089503036047489264611, 2.78198254310737045628424409153, 4.23698579650760407945864370138, 4.88609856105951209797534173292, 5.51618626742327860563072320775, 6.01221882342850302089762017150, 6.93681868022287711421127559494, 7.54245964266327663708826214166, 8.241502331889372565439388543213
