| L(s) = 1 | − 0.879·3-s + 0.694·5-s − 2.22·9-s − 4.36·11-s + 3.71·13-s − 0.610·15-s − 2.53·17-s − 2.29·19-s − 3.94·23-s − 4.51·25-s + 4.59·27-s − 0.241·29-s + 5.38·31-s + 3.84·33-s − 11.1·37-s − 3.26·39-s + 41-s + 4.98·43-s − 1.54·45-s − 9.92·47-s + 2.22·51-s + 12.8·53-s − 3.03·55-s + 2.01·57-s + 3.38·59-s + 6.24·61-s + 2.58·65-s + ⋯ |
| L(s) = 1 | − 0.507·3-s + 0.310·5-s − 0.742·9-s − 1.31·11-s + 1.03·13-s − 0.157·15-s − 0.614·17-s − 0.525·19-s − 0.822·23-s − 0.903·25-s + 0.884·27-s − 0.0447·29-s + 0.967·31-s + 0.668·33-s − 1.82·37-s − 0.523·39-s + 0.156·41-s + 0.760·43-s − 0.230·45-s − 1.44·47-s + 0.311·51-s + 1.76·53-s − 0.409·55-s + 0.266·57-s + 0.441·59-s + 0.799·61-s + 0.320·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9407782408\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9407782408\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 + 0.879T + 3T^{2} \) |
| 5 | \( 1 - 0.694T + 5T^{2} \) |
| 11 | \( 1 + 4.36T + 11T^{2} \) |
| 13 | \( 1 - 3.71T + 13T^{2} \) |
| 17 | \( 1 + 2.53T + 17T^{2} \) |
| 19 | \( 1 + 2.29T + 19T^{2} \) |
| 23 | \( 1 + 3.94T + 23T^{2} \) |
| 29 | \( 1 + 0.241T + 29T^{2} \) |
| 31 | \( 1 - 5.38T + 31T^{2} \) |
| 37 | \( 1 + 11.1T + 37T^{2} \) |
| 43 | \( 1 - 4.98T + 43T^{2} \) |
| 47 | \( 1 + 9.92T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 - 3.38T + 59T^{2} \) |
| 61 | \( 1 - 6.24T + 61T^{2} \) |
| 67 | \( 1 + 7.38T + 67T^{2} \) |
| 71 | \( 1 + 1.43T + 71T^{2} \) |
| 73 | \( 1 - 2.61T + 73T^{2} \) |
| 79 | \( 1 - 1.91T + 79T^{2} \) |
| 83 | \( 1 + 1.51T + 83T^{2} \) |
| 89 | \( 1 - 2.32T + 89T^{2} \) |
| 97 | \( 1 + 4.26T + 97T^{2} \) |
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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
−8.009957007242966894672526215241, −7.01680244584457631112103202618, −6.30065719283655566542788805565, −5.74833831602528269730088178406, −5.23200476854837271374092992326, −4.36465833139029406599992819016, −3.50082659386873857174520808414, −2.58763324687183055811830306112, −1.85677240800472376910598874370, −0.47039839218349477654940827214,
0.47039839218349477654940827214, 1.85677240800472376910598874370, 2.58763324687183055811830306112, 3.50082659386873857174520808414, 4.36465833139029406599992819016, 5.23200476854837271374092992326, 5.74833831602528269730088178406, 6.30065719283655566542788805565, 7.01680244584457631112103202618, 8.009957007242966894672526215241
