| L(s) = 1 | + 3.13·3-s + 3.22·5-s − 4.22·7-s + 6.80·9-s + 0.360·11-s + 0.769·13-s + 10.1·15-s − 0.871·17-s − 19-s − 13.2·21-s + 2.99·23-s + 5.41·25-s + 11.9·27-s − 0.0640·29-s + 10.6·31-s + 1.13·33-s − 13.6·35-s + 0.722·37-s + 2.40·39-s + 1.46·41-s + 0.946·43-s + 21.9·45-s + 3.57·47-s + 10.8·49-s − 2.72·51-s − 53-s + 1.16·55-s + ⋯ |
| L(s) = 1 | + 1.80·3-s + 1.44·5-s − 1.59·7-s + 2.26·9-s + 0.108·11-s + 0.213·13-s + 2.60·15-s − 0.211·17-s − 0.229·19-s − 2.88·21-s + 0.624·23-s + 1.08·25-s + 2.29·27-s − 0.0118·29-s + 1.90·31-s + 0.196·33-s − 2.30·35-s + 0.118·37-s + 0.385·39-s + 0.228·41-s + 0.144·43-s + 3.27·45-s + 0.521·47-s + 1.55·49-s − 0.382·51-s − 0.137·53-s + 0.157·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.486916837\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.486916837\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| good | 3 | \( 1 - 3.13T + 3T^{2} \) |
| 5 | \( 1 - 3.22T + 5T^{2} \) |
| 7 | \( 1 + 4.22T + 7T^{2} \) |
| 11 | \( 1 - 0.360T + 11T^{2} \) |
| 13 | \( 1 - 0.769T + 13T^{2} \) |
| 17 | \( 1 + 0.871T + 17T^{2} \) |
| 23 | \( 1 - 2.99T + 23T^{2} \) |
| 29 | \( 1 + 0.0640T + 29T^{2} \) |
| 31 | \( 1 - 10.6T + 31T^{2} \) |
| 37 | \( 1 - 0.722T + 37T^{2} \) |
| 41 | \( 1 - 1.46T + 41T^{2} \) |
| 43 | \( 1 - 0.946T + 43T^{2} \) |
| 47 | \( 1 - 3.57T + 47T^{2} \) |
| 59 | \( 1 + 9.43T + 59T^{2} \) |
| 61 | \( 1 - 3.77T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 + 6.74T + 73T^{2} \) |
| 79 | \( 1 + 3.97T + 79T^{2} \) |
| 83 | \( 1 - 9.49T + 83T^{2} \) |
| 89 | \( 1 + 7.31T + 89T^{2} \) |
| 97 | \( 1 + 18.3T + 97T^{2} \) |
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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
−8.629584105076052694489667950403, −7.889752937347524198172230908550, −6.76454743864567847478025103196, −6.56546326244934236829715304278, −5.60219086232475641921695113911, −4.42629345396220807905613390184, −3.54773160320921927208757472927, −2.77038894338688831056303208459, −2.35992722322895330652569604361, −1.19960008669630531372301603503,
1.19960008669630531372301603503, 2.35992722322895330652569604361, 2.77038894338688831056303208459, 3.54773160320921927208757472927, 4.42629345396220807905613390184, 5.60219086232475641921695113911, 6.56546326244934236829715304278, 6.76454743864567847478025103196, 7.889752937347524198172230908550, 8.629584105076052694489667950403
