| L(s) = 1 | + 0.198·3-s + 3.93·5-s − 2.96·9-s + 11-s − 0.692·13-s + 0.780·15-s − 0.850·17-s − 0.753·19-s − 0.643·23-s + 10.5·25-s − 1.18·27-s + 5.43·29-s − 6.44·31-s + 0.198·33-s + 10.0·37-s − 0.137·39-s − 1.89·41-s + 2.97·43-s − 11.6·45-s + 12.6·47-s − 0.168·51-s − 2.02·53-s + 3.93·55-s − 0.149·57-s + 7.73·59-s + 9.48·61-s − 2.72·65-s + ⋯ |
| L(s) = 1 | + 0.114·3-s + 1.76·5-s − 0.986·9-s + 0.301·11-s − 0.191·13-s + 0.201·15-s − 0.206·17-s − 0.172·19-s − 0.134·23-s + 2.10·25-s − 0.227·27-s + 1.00·29-s − 1.15·31-s + 0.0344·33-s + 1.66·37-s − 0.0219·39-s − 0.296·41-s + 0.454·43-s − 1.73·45-s + 1.84·47-s − 0.0235·51-s − 0.277·53-s + 0.531·55-s − 0.0197·57-s + 1.00·59-s + 1.21·61-s − 0.338·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.723817287\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.723817287\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 0.198T + 3T^{2} \) |
| 5 | \( 1 - 3.93T + 5T^{2} \) |
| 13 | \( 1 + 0.692T + 13T^{2} \) |
| 17 | \( 1 + 0.850T + 17T^{2} \) |
| 19 | \( 1 + 0.753T + 19T^{2} \) |
| 23 | \( 1 + 0.643T + 23T^{2} \) |
| 29 | \( 1 - 5.43T + 29T^{2} \) |
| 31 | \( 1 + 6.44T + 31T^{2} \) |
| 37 | \( 1 - 10.0T + 37T^{2} \) |
| 41 | \( 1 + 1.89T + 41T^{2} \) |
| 43 | \( 1 - 2.97T + 43T^{2} \) |
| 47 | \( 1 - 12.6T + 47T^{2} \) |
| 53 | \( 1 + 2.02T + 53T^{2} \) |
| 59 | \( 1 - 7.73T + 59T^{2} \) |
| 61 | \( 1 - 9.48T + 61T^{2} \) |
| 67 | \( 1 + 8.81T + 67T^{2} \) |
| 71 | \( 1 + 0.313T + 71T^{2} \) |
| 73 | \( 1 - 16.0T + 73T^{2} \) |
| 79 | \( 1 - 7.55T + 79T^{2} \) |
| 83 | \( 1 - 13.1T + 83T^{2} \) |
| 89 | \( 1 + 1.06T + 89T^{2} \) |
| 97 | \( 1 + 5.85T + 97T^{2} \) |
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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
−8.594490598552694299705044100222, −7.67997593459492388981815390415, −6.68974478404802859242495649465, −6.12370129393072267450977788395, −5.55231616172210074134360108817, −4.85830889519538589957149414243, −3.72762225945860069274362833431, −2.58092085877129103351773440293, −2.19433629158390195156756478506, −0.940295833446548825396430242538,
0.940295833446548825396430242538, 2.19433629158390195156756478506, 2.58092085877129103351773440293, 3.72762225945860069274362833431, 4.85830889519538589957149414243, 5.55231616172210074134360108817, 6.12370129393072267450977788395, 6.68974478404802859242495649465, 7.67997593459492388981815390415, 8.594490598552694299705044100222
