| L(s) = 1 | + 3-s + 5-s − 2.61·7-s + 9-s − 2.61·11-s − 13-s + 15-s + 4.39·17-s − 7.77·19-s − 2.61·21-s + 6.39·23-s + 25-s + 27-s + 3.23·29-s − 5.00·31-s − 2.61·33-s − 2.61·35-s − 4.39·37-s − 39-s + 11.1·41-s − 1.23·43-s + 45-s + 9.00·47-s − 0.161·49-s + 4.39·51-s + 12.3·53-s − 2.61·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.988·7-s + 0.333·9-s − 0.788·11-s − 0.277·13-s + 0.258·15-s + 1.06·17-s − 1.78·19-s − 0.570·21-s + 1.33·23-s + 0.200·25-s + 0.192·27-s + 0.599·29-s − 0.899·31-s − 0.455·33-s − 0.442·35-s − 0.721·37-s − 0.160·39-s + 1.74·41-s − 0.187·43-s + 0.149·45-s + 1.31·47-s − 0.0230·49-s + 0.614·51-s + 1.70·53-s − 0.352·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.118540393\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.118540393\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 2.61T + 7T^{2} \) |
| 11 | \( 1 + 2.61T + 11T^{2} \) |
| 17 | \( 1 - 4.39T + 17T^{2} \) |
| 19 | \( 1 + 7.77T + 19T^{2} \) |
| 23 | \( 1 - 6.39T + 23T^{2} \) |
| 29 | \( 1 - 3.23T + 29T^{2} \) |
| 31 | \( 1 + 5.00T + 31T^{2} \) |
| 37 | \( 1 + 4.39T + 37T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 + 1.23T + 43T^{2} \) |
| 47 | \( 1 - 9.00T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 + 9.00T + 59T^{2} \) |
| 61 | \( 1 + 0.391T + 61T^{2} \) |
| 67 | \( 1 - 13.0T + 67T^{2} \) |
| 71 | \( 1 - 5.38T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 + 1.62T + 89T^{2} \) |
| 97 | \( 1 - 5.93T + 97T^{2} \) |
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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
−8.079706760413195092725749658180, −7.29646779196397506999135173318, −6.73408704127153524790192689211, −5.92073156550247677917333959226, −5.26677054310896890789210450619, −4.34302775942617850692286532145, −3.47246900530575828054085729581, −2.74498081895264370434857311334, −2.08909182150442623991551336415, −0.71604970091783820723661542809,
0.71604970091783820723661542809, 2.08909182150442623991551336415, 2.74498081895264370434857311334, 3.47246900530575828054085729581, 4.34302775942617850692286532145, 5.26677054310896890789210450619, 5.92073156550247677917333959226, 6.73408704127153524790192689211, 7.29646779196397506999135173318, 8.079706760413195092725749658180
